Unique prime

Unique primeNo. of known terms| 102

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Conjectured no. of terms| Infinite
First terms| 3, 11, 37, 101
Largest known term| (10270343-1)/9
OEIS index|

* A040017
* Unique period primes (no other prime has same period as 1/p) in order (periods are given in A051627)

In recreational number theory, a unique prime or unique period prime is a certain kind of prime number. A prime p ≠ 2, 5 is called unique if there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1 / p, is equal to the period length of the reciprocal of q, 1 / q.[1] For example, 3 is the only prime with period 1, 11 is the only prime with period 2, 37 is the only prime with period 3, 101 is the only prime with period 4, so they are unique primes. In contrast, 41 and 271 both have period 5; 7 and 13 both have period 6; 239 and 4649 both have period 7; 73 and 137 both have period 8; 21649 and 513239 both have period 11; 53, 79 and 265371653 all have period 13; 31 and 2906161 both have period 15; 17 and 5882353 both have period 16; 2071723 and 5363222357 both have period 17; 19 and 52579 both have period 18; 3541 and 27961 both have period 20. Therefore, none of these is a unique prime. Unique primes were first described by Samuel Yates in 1980.

The above definition is related to the decimal representation of integers. Unique primes may be defined and have been studied in any numeral base.

## Contents

* 1 Period of a prime in base b
* 2 Decimal unique primes
* 3 Binary unique primes
* 4 Bi-unique primes
* 5 Period lengths
* 6 Unique prime in various bases
* 7 Bibliography
* 8 References
* 9 External links

## Period of a prime in base b

The representation of the reciprocal of a prime number (or, more generally, an integer) p in the numeral base b is periodic of period n if

[math]\displaystyle{ \frac{1}{p}=\sum_{i=1}^\infty \frac{q}{(b^n)^i}, }[/math]

where q is a positive integer smaller than [math]\displaystyle{ b^n. }[/math] According to the summation formula of geometric series, this may be rewritten as

[math]\displaystyle{ \frac{1}{p}=\frac{q}{b^n-1}. }[/math]

In other words, n is a period of the representation of 1/p if and only if p is a divisor of [math]\displaystyle{ b^n-1. }[/math] Euler's theorem asserts that, if an integer b is coprime with p, then p is a divisor of [math]\displaystyle{ p^{\varphi(n)}-1 }[/math] where [math]\displaystyle{ \varphi }[/math] is Euler's totient function. This proves that, for every integer p coprime with b, the representation of the reciprocal of p is periodic in base b.

All the periods of a periodic function are multiples of a shortest period generally called the fundamental period. In this article, we call period of p in base b the shortest period of the representation of 1/p in base b. Therefore, the period of p in base b is the smallest positive integer n such that that p is a divisor of [math]\displaystyle{ b^n-1. }[/math] In other words, the period of a prime p in base b is the multiplicative order of b modulo p.

According to Zsigmondy's theorem, every positive integer is a period of some prime in base b except in the following cases:

* b = 2 and n = 1 or 6
* n = 2 and b= 2k − 1 for some integer k > 1

[math]\displaystyle{ x^n-1=\prod_{i\mid n} \Phi_n(x), }[/math]

where [math]\displaystyle{ \Phi_n }[/math] is the nth cyclotomic polynomial, the primes of period n in base b are prime divisors of [math]\displaystyle{ \Phi_n(b). }[/math] More precisely, the primes of period n are exactly the prime divisors of [math]\displaystyle{ \Phi_n(b) }[/math] that do not divide n (see below for a proof of this result and of the following ones).

If b is even (this includes the binary and the decimal cases), the prime divisors of [math]\displaystyle{ \Phi_n(b) }[/math] that do not divide n are exactly the prime divisors of[citation needed]

[math]\displaystyle{ R_n(b)=\frac{\Phi_n(b)}{\gcd(\Phi_n(b),n)}. }[/math]

This is wrong if b is odd: if n = 2 and b = 4k − 1, where k is a positive integer, then

[math]\displaystyle{ R_2(b)=\frac{\Phi_2(b)}{\gcd(\Phi_2(b), 2)} = \frac{b+1}{2} = 2k, }[/math]

although 2 divides both n = 2 and [math]\displaystyle{ \Phi_n(b). }[/math]

If b is odd, the primes of period n are exactly, if n = 1, the prime divisors of [math]\displaystyle{ R_1(b) = b-1 }[/math], or, if n > 1, the odd prime divisors of Rn(b).

Sketch of the proof of the characterization of primes of period n

As the period of every prime p divides p – 1 (Fermat's little theorem), if p divides n, then its period is smaller than n. Conversely, if p divides [math]\displaystyle{ \Phi_n(b) }[/math] and has a period k smaller than n, then it is a common divisor of [math]\displaystyle{ \Phi_n(b) }[/math] and [math]\displaystyle{ \Phi_k(b). }[/math] As the resultant of two polynomials is a linear combination of these polynomials, p divides the resultant of [math]\displaystyle{ \Phi_n(x) }[/math] and [math]\displaystyle{ \Phi_k(x). }[/math] As these two polynomials are coprime and divide [math]\displaystyle{ x^n-1, }[/math] p divides also the discriminant [math]\displaystyle{ n^n }[/math] of [math]\displaystyle{ x^n-1. }[/math] Thus, a prime divisor of [math]\displaystyle{ \Phi_n(b) }[/math], that has a period smaller than n, is also a divisor of n.

Now, we have to prove that, if a prime p > 2 divides n and [math]\displaystyle{ \Phi_n(b), }[/math] then it does not divide [math]\displaystyle{ \Phi_n(b)/p. }[/math] In fact, this implies immediately that p does not divide [math]\displaystyle{ \Phi_n(b)/\gcd(\Phi_n(b),n). }[/math] If b is even, 2 cannot divide [math]\displaystyle{ \Phi_n(b), }[/math] (which is odd), and the condition p > 2 is not restrictive.

Thus, let n = pm. It suffices to prove that [math]\displaystyle{ p^2 }[/math] does not divides S(b) for some polynomial S(x), which is a multiple of [math]\displaystyle{ \Phi_n(x). }[/math] We take

[math]\displaystyle{ S(x)=\frac{x^n-1}{x^m-1} = 1 + x^m + x^{2m} + \cdots + x^{(p-1)m}. }[/math]

By Fermat's little theorem, we have [math]\displaystyle{ b^{p-1}\equiv 1 \pmod p. }[/math] As p divides [math]\displaystyle{ \Phi_n(b) }[/math], we have also [math]\displaystyle{ b^{n}\equiv 1 \pmod p. }[/math] Thus the multiplicative order of b modulo p divides gcd(n, p − 1), which is a divisor of m = n/p. Thus c = bm − 1 is a multiple of p. Now,

[math]\displaystyle{ S(b) = \frac{(1+c)^p-1}{c} = p+ \binom{p}{2}c + \cdots + \binom{p}{p}c^{p-1}. }[/math]

As p is prime and greater than 2, all the terms but the first one are multiple of [math]\displaystyle{ p^2. }[/math] This proves that [math]\displaystyle{ p^2 }[/math] does not divides [math]\displaystyle{ \Phi_n(b). }[/math]

A prime p is a unique prime in base b, if and only if, for some n, it is the unique prime divisor of [math]\displaystyle{ \Phi_n(b) }[/math] that does not divide n. If b is even (which includes the binary and the decimal cases) this means that

[math]\displaystyle{ R_n(b)=\frac{\Phi_n(b)}{\gcd(\Phi_n(b),n)} = p^c. }[/math]

for some positive integer c .

If b is odd, this means that

[math]\displaystyle{ R_n(b)=\frac{\Phi_n(b)}{\gcd(\Phi_n(b),n)} =p^c 2^d. }[/math]

for some integers c > 0 and d ≥ 0. This provides an efficient method for computing the unique primes and the primes of a given period.

Note that a prime divisor of b is coprime with [math]\displaystyle{ b^n-1 }[/math], and thus also with its divisor [math]\displaystyle{ \Phi_n(b). }[/math] Such a prime has no period length, as the representation in base b of its reciprocal is finite instead of being periodic. Thus, such a prime is never considered as a unique prime, even if it is the unique prime that has a finite reciprocal in base b. For example, 2 is not considered as a unique prime in binary, although it is the only prime with finite reciprocal in binary.

There are four situations of unique prime in base b:

1. [math]\displaystyle{ \Phi_n(b)=p }[/math] and p is prime.
2. [math]\displaystyle{ \frac{\Phi_n(b)}{\gcd(\Phi_n(b),n)}=p }[/math] and p is prime and [math]\displaystyle{ \gcd(\Phi_n(b),n)\gt 1 }[/math].
3. [math]\displaystyle{ \Phi_n(b)=p^c }[/math] and p is prime and [math]\displaystyle{ c\gt 1 }[/math].
4. [math]\displaystyle{ \frac{\Phi_n(b)}{\gcd(\Phi_n(b),n)}=p^c }[/math] and p is prime and [math]\displaystyle{ \gcd(\Phi_n(b),n)\gt 1 }[/math] and [math]\displaystyle{ c\gt 1 }[/math].

Conjectures:

* For all positive integers n, there are infinitely many bases b such that there is a unique prime with period length n. (this conjecture is true if Bunyakowsky conjecture is true, since cyclotomic polynomials are irreducible)

* For all bases b, there are infinitely many unique primes.

* For all positive integers n, there are infinitely many bases b such that there is a unique prime with period length n which is the 1st situation. (i.e. for all n≥1, there are infinitely many b≥2 such that [math]\displaystyle{ \Phi_n(b) }[/math] is prime, where [math]\displaystyle{ \Phi }[/math] is the cyclotomic polynomial) (this conjecture is true if Bunyakowsky conjecture is true, since cyclotomic polynomials are irreducible)

* For all bases b, there are infinitely many unique primes which are the 1st situation. (i.e. for all b≥2, there are infinitely many n≥1 such that [math]\displaystyle{ \Phi_n(b) }[/math] is prime, where [math]\displaystyle{ \Phi }[/math] is the cyclotomic polynomial)

* For all positive integers n not in (sequence A253235 in the OEIS), there are infinitely many bases b such that there is a unique prime with period length n which is the 2nd situation. (periods lengths n in (sequence A253235 in the OEIS) cannot have any bases which is the 2nd or the 4th situation) (this conjecture is true if Bunyakowsky conjecture is true, since cyclotomic polynomials are irreducible)

* For all bases b, there are only finitely many unique primes (or no unique primes) which are the 2nd situation, specially, in base 2, the only known periods which are the 2nd situation are 18, 20, 21, 54, 147, 342, 602, 889, 258121, and in base 10, the only known periods which are the 2nd situation is 3, 9, 294.

* All (period n, base b) combos which is the 3rd situation are

*     * n = 1, b = pc+1 with p prime and c ≥ 2
* n = 2, b = pc−1 with p odd prime and c ≥ 2
* n = 3, b = 18
* n = 5, b = 3
* n = 6, b = 19

* All (period n, base b) combos which is the 4th situation are

*     * n = 2, b = 2rpc−1 with p odd prime and r ≥ 1 and c ≥ 2
* n = 3, b in (sequence A028231 in the OEIS) in the case of (b2+b+1)/3 is square of prime (it is conjectured that powers with exponent > 2 cannot exist) (the first such bases b are 22, 313, 4366, 847078, ...) (correspond prime are in (sequence A001570 in the OEIS))
* n = 4, b in (sequence A002315 in the OEIS) in the case of (b2+1)/2 is square of prime or 28561=134 (it is conjectured that powers with exponent > 2 cannot exist, except (2392+1)/2 = 28561 = 134) (the first such bases b are 7, 41, 239, 8119, 47321, ...) (correspond prime are in (sequence A001563 in the OEIS))
* n = 6, b−1 in (sequence A028231 in the OEIS) in the case of (b2−b+1)/3 is square of prime (it is conjectured that powers with exponent > 2 cannot exist) (the first such bases b are 23, 314, 4367, 847079, ...) (correspond prime are in (sequence A001570 in the OEIS))

* Specially, no (period n, base b) combo which is the 3rd or the 4th situation exist for n > 6

* “Almost all” (period n, base b) combos are the 1st situation

(For all (period n, base b) combo which is the 3rd or the 4th situation, the corresponding prime p must be generalized Wieferich prime to base b)

Table of the periods of primes up to 139 in bases up to 24

The mention "terminated" means that the prime divides the base, and thus that the representation of its reciprocal is finite.

base

prime

| 2 | 3  | 4  | 5  | 6  | 7  | 8  | 9  | 10  | 11  | 12  | 13  | 14  | 15  | 16  | 17  | 18  | 19  | 20  | 21  | 22  | 23  | 24

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2  | Terminated  | 1  | Terminated  | 1  | Terminated  | 1  | Terminated  | 1  | Terminated  | 1  | Terminated  | 1  | Terminated  | 1  | Terminated  | 1  | Terminated  | 1  | Terminated  | 1  | Terminated  | 1  | Terminated
3  | 2  | Terminated  | 1  | 2  | Terminated  | 1  | 2  | Terminated  | 1  | 2  | Terminated  | 1  | 2  | Terminated  | 1  | 2  | Terminated  | 1  | 2  | Terminated  | 1  | 2  | Terminated
5  | 4  | 4  | 2  | Terminated  | 1  | 4  | 4  | 2  | Terminated  | 1  | 4  | 4  | 2  | Terminated  | 1  | 4  | 4  | 2  | Terminated  | 1  | 4  | 4  | 2
7  | 3  | 6  | 3  | 6  | 2  | Terminated  | 1  | 3  | 6  | 3  | 6  | 2  | Terminated  | 1  | 3  | 6  | 3  | 6  | 2  | Terminated  | 1  | 3  | 6
11  | 10  | 5  | 5  | 5  | 10  | 10  | 10  | 5  | 2  | Terminated  | 1  | 10  | 5  | 5  | 5  | 10  | 10  | 10  | 5  | 2  | Terminated  | 1  | 10
13  | 12  | 3  | 6  | 4  | 12  | 12  | 4  | 3  | 6  | 12  | 2  | Terminated  | 1  | 12  | 3  | 6  | 4  | 12  | 12  | 4  | 3  | 6  | 12
17  | 8  | 16  | 4  | 16  | 16  | 16  | 8  | 8  | 16  | 16  | 16  | 4  | 16  | 8  | 2  | Terminated  | 1  | 8  | 16  | 4  | 16  | 16  | 16
19  | 18  | 18  | 9  | 9  | 9  | 3  | 6  | 9  | 18  | 3  | 6  | 18  | 18  | 18  | 9  | 9  | 2  | Terminated  | 1  | 18  | 18  | 9  | 9
23  | 11  | 11  | 11  | 22  | 11  | 22  | 11  | 11  | 22  | 22  | 11  | 11  | 22  | 22  | 11  | 22  | 11  | 22  | 22  | 22  | 2  | Terminated  | 1
29  | 28  | 28  | 14  | 14  | 14  | 7  | 28  | 14  | 28  | 28  | 4  | 14  | 28  | 28  | 7  | 4  | 28  | 28  | 7  | 28  | 14  | 7  | 7
31  | 5  | 30  | 5  | 3  | 6  | 15  | 5  | 15  | 15  | 30  | 30  | 30  | 15  | 10  | 5  | 30  | 15  | 15  | 15  | 30  | 30  | 10  | 30
37  | 36  | 18  | 18  | 36  | 4  | 9  | 12  | 9  | 3  | 6  | 9  | 36  | 12  | 36  | 9  | 36  | 36  | 36  | 36  | 18  | 36  | 12  | 36
41  | 20  | 8  | 10  | 20  | 40  | 40  | 20  | 4  | 5  | 40  | 40  | 40  | 8  | 40  | 5  | 40  | 5  | 40  | 20  | 20  | 40  | 10  | 40
43  | 14  | 42  | 7  | 42  | 3  | 6  | 14  | 21  | 21  | 7  | 42  | 21  | 21  | 21  | 7  | 21  | 42  | 42  | 42  | 7  | 14  | 21  | 21
47  | 23  | 23  | 23  | 46  | 23  | 23  | 23  | 23  | 46  | 46  | 23  | 46  | 23  | 46  | 23  | 23  | 23  | 46  | 46  | 23  | 46  | 46  | 23
53  | 52  | 52  | 26  | 52  | 26  | 26  | 52  | 26  | 13  | 26  | 52  | 13  | 52  | 13  | 13  | 26  | 52  | 52  | 52  | 52  | 52  | 4  | 13
59  | 58  | 29  | 29  | 29  | 58  | 29  | 58  | 29  | 58  | 58  | 29  | 58  | 58  | 29  | 29  | 29  | 58  | 29  | 29  | 29  | 29  | 58  | 58
61  | 60  | 10  | 30  | 30  | 60  | 60  | 20  | 5  | 60  | 4  | 15  | 3  | 6  | 15  | 15  | 60  | 60  | 30  | 5  | 12  | 15  | 20  | 20
67  | 66  | 22  | 33  | 22  | 33  | 66  | 22  | 11  | 33  | 66  | 66  | 66  | 11  | 11  | 33  | 33  | 66  | 33  | 66  | 33  | 11  | 33  | 11
71  | 35  | 35  | 35  | 5  | 35  | 70  | 35  | 35  | 35  | 70  | 35  | 70  | 10  | 35  | 35  | 10  | 35  | 35  | 7  | 70  | 70  | 14  | 35
73  | 9  | 12  | 9  | 72  | 36  | 24  | 3  | 6  | 8  | 72  | 36  | 72  | 72  | 72  | 9  | 24  | 18  | 36  | 72  | 24  | 8  | 36  | 12
79  | 39  | 78  | 39  | 39  | 78  | 78  | 13  | 39  | 13  | 39  | 26  | 39  | 26  | 26  | 39  | 26  | 13  | 39  | 39  | 13  | 13  | 3  | 6
83  | 82  | 41  | 41  | 82  | 82  | 41  | 82  | 41  | 41  | 41  | 41  | 82  | 82  | 82  | 41  | 41  | 82  | 82  | 82  | 41  | 82  | 41  | 82
89  | 11  | 88  | 11  | 44  | 88  | 88  | 11  | 44  | 44  | 22  | 8  | 88  | 88  | 88  | 11  | 44  | 44  | 88  | 44  | 44  | 22  | 88  | 88
97  | 48  | 48  | 24  | 96  | 12  | 96  | 16  | 24  | 96  | 48  | 16  | 96  | 96  | 96  | 12  | 96  | 16  | 32  | 32  | 96  | 4  | 96  | 24
101  | 100  | 100  | 50  | 25  | 10  | 100  | 100  | 50  | 4  | 100  | 100  | 50  | 10  | 100  | 25  | 10  | 100  | 25  | 50  | 50  | 50  | 50  | 25
103  | 51  | 34  | 51  | 102  | 102  | 51  | 17  | 17  | 34  | 102  | 102  | 17  | 17  | 51  | 51  | 51  | 51  | 51  | 102  | 102  | 34  | 17  | 34
107  | 106  | 53  | 53  | 106  | 106  | 106  | 106  | 53  | 53  | 53  | 53  | 53  | 53  | 106  | 53  | 106  | 106  | 53  | 106  | 106  | 106  | 53  | 106
109  | 36  | 27  | 18  | 27  | 108  | 27  | 12  | 27  | 108  | 108  | 54  | 108  | 108  | 27  | 9  | 36  | 108  | 36  | 54  | 27  | 27  | 36  | 108
113  | 28  | 112  | 14  | 112  | 112  | 14  | 28  | 56  | 112  | 56  | 112  | 56  | 28  | 4  | 7  | 112  | 8  | 112  | 112  | 112  | 56  | 112  | 112
127  | 7  | 126  | 7  | 42  | 126  | 126  | 7  | 63  | 42  | 63  | 126  | 63  | 126  | 63  | 7  | 63  | 63  | 3  | 6  | 63  | 9  | 126  | 18
131  | 130  | 65  | 65  | 65  | 130  | 65  | 130  | 65  | 130  | 65  | 65  | 65  | 130  | 65  | 65  | 130  | 26  | 26  | 65  | 65  | 130  | 130  | 26
137  | 68  | 136  | 34  | 136  | 136  | 68  | 68  | 68  | 8  | 68  | 136  | 136  | 34  | 34  | 17  | 68  | 34  | 68  | 136  | 136  | 34  | 136  | 136
139  | 138  | 138  | 69  | 69  | 23  | 69  | 46  | 69  | 46  | 69  | 138  | 69  | 46  | 138  | 69  | 138  | 138  | 138  | 69  | 138  | 138  | 46  | 69

Table of primes of a given period (up to 24) in bases up to 24

Bold for unique primes.

base

period
length

| 2 | 3  | 4  | 5  | 6  | 7  | 8  | 9  | 10  | 11  | 12  | 13  | 14  | 15  | 16  | 17  | 18  | 19  | 20  | 21  | 22  | 23  | 24

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1  | (none)  | 2 | 3 | 2 | 5 | 2, 3  | 7 | 2 | 3 | 2, 5  | 11 | 2, 3  | 13 | 2, 7  | 3, 5  | 2 | 17 | 2, 3  | 19 | 2, 5  | 3, 7  | 2, 11  | 23
2  | 3 | (none)  | 5 | 3 | 7 | (none)  | 3 | 5 | 11 | 3 | 13 | 7 | 3, 5  | (none)  | 17 | 3 | 19 | 5 | 3, 7  | 11 | 23 | 3 | 5
3  | 7 | 13 | 7 | 31 | 43 | 19 | 73 | 7, 13  | 37 | 7, 19  | 157 | 61 | 211 | 241 | 7, 13  | 307 | 7 | 127 | 421 | 463 | 13 | 7, 79  | 601
4  | 5 | 5 | 17 | 13 | 37 | 5 | 5, 13  | 41 | 101 | 61 | 5, 29  | 5, 17  | 197 | 113 | 257 | 5, 29  | 5, 13  | 181 | 401 | 13, 17  | 5, 97  | 5, 53  | 577
5  | 31 | 11 | 11, 31  | 11, 71  | 311 | 2801 | 31, 151  | 11, 61  | 41, 271  | 3221 | 22621 | 30941 | 11, 3761  | 11, 4931  | 11, 31, 41  | 88741 | 41, 2711  | 151, 911  | 11, 61, 251  | 40841 | 245411 | 292561 | 346201
6  | (none)  | 7 | 13 | 7 | 31 | 43 | 19 | 73 | 7, 13  | 37 | 7, 19  | 157 | 61 | 211 | 241 | 7, 13  | 307 | 7 | 127 | 421 | 463 | 13 | 7, 79
7  | 127 | 1093 | 43, 127  | 19531 | 55987 | 29, 4733  | 127, 337  | 547, 1093  | 239, 4649  | 43, 45319  | 659, 4943  | 5229043 | 8108731 | 1743463 | 29, 43, 113, 127  | 25646167 | 449, 80207  | 701, 70841  | 29, 71, 32719  | 43, 631, 3319  | 16968421 | 29, 5336717  | 29, 239, 28771
8  | 17 | 41 | 257 | 313 | 1297 | 1201 | 17, 241  | 17, 193  | 73, 137  | 7321 | 89, 233  | 14281 | 41, 937  | 17, 1489  | 65537 | 41761 | 113, 929  | 17, 3833  | 160001 | 97241 | 73, 3209  | 139921 | 331777
9  | 73 | 757 | 19, 73  | 19, 829  | 19, 2467  | 37, 1063  | 262657 | 19, 37, 757  | 333667 | 1772893 | 37, 80749  | 1609669 | 397, 18973  | 541, 21061  | 19, 37, 73, 109  | 19, 1270657  | 991, 34327  | 523, 29989  | 64008001 | 85775383 | 127, 297613  | 19, 7792003  | 19, 2017, 4987
10  | 11 | 61 | 41 | 521 | 11, 101  | 11, 191  | 11, 331  | 1181 | 9091 | 13421 | 19141 | 11, 2411  | 71, 101  | 31, 1531  | 61681 | 11, 71, 101  | 11, 9041  | 11, 2251  | 152381 | 185641 | 224071 | 31, 41, 211  | 11, 5791
11  | 23, 89  | 23, 3851  | 23, 89, 683  | 12207031 | 23, 3154757  | 1123, 293459  | 23, 89, 599479  | 23, 67, 661, 3851  | 21649, 513239  | 15797, 1806113  | 23, 266981089  | 23, 419, 859, 18041  | 67, 4027, 1154539  | 67, 463, 2333, 8537  | 23, 89, 397, 683, 2113  | 2141993519227 | 23, 199, 16127, 51217  | 104281, 62060021  | 10778947368421 | 17513875027111 | 67, 353, 1176469537  | 3937230404603 | 67, 7349, 134367047
12  | 13 | 73 | 241 | 601 | 13, 97  | 13, 181  | 37, 109  | 6481 | 9901 | 13, 1117  | 20593 | 28393 | 37, 1033  | 13, 3877  | 97, 673  | 83233 | 229, 457  | 13, 769  | 13, 12277  | 61, 3181  | 157, 1489  | 37, 7549  | 13, 73, 349
13  | 8191 | 797161 | 2731, 8191  | 305175781 | 3433, 760891  | 16148168401 | 79, 8191, 121369  | 398581, 797161  | 53, 79, 265371653  | 1093, 3158528101  | 477517, 20369233  | 53, 264031, 1803647  | 157, 29914249171  | 53, 157483, 16655159  | 53, 157, 1613, 2731, 8191  | 212057, 2919196853  | 79, 521, 29759719289  | 599, 29251, 133338869  | 3121, 142559, 9690539  | 79, 189437, 516094151  | 79, 2003, 85107437663  | 47691619, 480393499  | 53, 6553, 15913, 6895253
14  | 43 | 547 | 29, 113  | 29, 449  | 29, 197  | 113, 911  | 43, 5419  | 29, 16493  | 909091 | 1623931 | 211, 13063  | 29, 22079  | 7027567 | 10678711 | 15790321 | 22796593 | 32222107 | 197, 226871  | 827, 10529  | 81867661 | 29, 43, 86969  | 71, 673, 2969  | 183458857
15  | 151 | 4561 | 151, 331  | 181, 1741  | 1171, 1201  | 31, 159871  | 631, 23311  | 31, 271, 4561  | 31, 2906161  | 195019441 | 61, 661, 9781  | 4651, 161971  | 31, 2851, 15511  | 61, 39225301  | 61, 151, 331, 1321  | 6566760001 | 31, 601, 558721  | 31, 211, 2460181  | 31, 3001, 261451  | 211, 9391, 18181  | 61, 858794191  | 74912328481 | 241, 17881, 24481
16  | 257 | 17, 193  | 65537 | 17, 11489  | 17, 98801  | 17, 169553  | 97, 257, 673  | 21523361 | 17, 5882353  | 17, 6304673  | 17, 97, 260753  | 407865361 | 17, 5393, 16097  | 7121, 179953  | 641, 6700417  | 18913, 184417  | 97, 113607841  | 15073, 563377  | 17, 1505882353  | 62897, 300673  | 17, 3227992561  | 17, 3697, 623009  | 17, 2801, 2311681
17  | 131071 | 1871, 34511  | 43691, 131071  | 409, 466344409  | 239, 409, 1123, 30839  | 14009, 2767631689  | 103, 2143, 11119, 131071  | 103, 307, 1021, 1871, 34511  | 2071723, 5363222357  | 50544702849929377 | 2693651, 74876782031  | 103, 443, 15798461357509  | 103, 22771730193675277  | 1045002649, 6734509609  | 137, 953, 26317, 43691, 131071  | 10949, 1749233, 2699538733  | 7563707819165039903 | 3044803, 99995282631947  | 689852631578947368421 | 1502097124754084594737 | 239, 74729519, 176634767651  | 103, 62246266355102810647  | 307, 120574031, 341563234253
18  | 19 | 19, 37  | 37, 109  | 5167 | 46441 | 117307 | 87211 | 530713 | 19, 52579  | 590077 | 1657, 1801  | 19, 271, 937  | 19, 132049  | 19, 739, 811  | 433, 38737  | 1423, 5653  | 73, 465841  | 199, 236377  | 307, 69481  | 19, 37, 199, 613  | 19, 5966803  | 163, 271, 1117  | 127, 199, 7561
19  | 524287 | 1597, 363889  | 174763, 524287  | 191, 6271, 3981071  | 191, 638073026189  | 419, 4534166740403  | 32377, 524287, 1212847  | 1597, 2851, 101917, 363889  | 1111111111111111111 | 6115909044841454629 | 29043636306420266077 | 12865927, 9468940004449  | 459715689149916492091 | 4272113, 370649274902657  | 229, 457, 174763, 524287, 525313  | 229, 1103, 202607147, 291973723  | 6841, 6089884909802812423  | 109912203092239643840221 | 75368484119, 192696104561  | 12061389013, 54921106624003  | 45943, 341203, 97404596002423  | 2129, 63877469, 24939218613613  | 7282588256957615350925401
20  | 41 | 1181 | 61681 | 41, 9161  | 241, 6781  | 281, 4021  | 41, 61, 1321  | 42521761 | 3541, 27961  | 212601841 | 85403261 | 421, 601, 641  | 1061, 1383881  | 19421, 131381  | 4278255361 | 21881, 63541  | 15101, 145501  | 16936647121 | 41, 2801, 222361  | 41, 920421641  | 181, 401, 150901  | 61, 941, 272341  | 61, 1801385941
21  | 337 | 368089 | 337, 5419  | 379, 519499  | 1822428931 | 11898664849 | 92737, 649657  | 43, 2269, 368089  | 43, 1933, 10838689  | 1723, 8527, 27763  | 8177824843189 | 43, 337, 547, 2714377  | 43, 547, 2239000891  | 43, 2817034275427  | 337, 1429, 5419, 14449  | 43, 13567, 940143709  | 156107192084257 | 30640261, 68443621  | 460951, 8442733531  | 4789, 6427, 227633407  | 12271836836138419 | 43, 170689, 408030421  | 43, 10426753, 78066619
22  | 683 | 67, 661  | 397, 2113  | 23, 67, 5281  | 51828151 | 23, 10746341  | 67, 683, 20857  | 5501, 570461  | 23, 4093, 8779  | 23, 89, 199, 58367  | 57154490053 | 128011456717 | 23, 11737870057  | 23, 23504771357  | 353, 2931542417  | 23, 947, 87415373  | 536801, 6301307  | 23, 253239693257  | 23, 424016563147  | 23, 6073, 10362529  | 89, 285451051007  | 39700406579747 | 60867245726761
23  | 47, 178481  | 47, 1001523179  | 47, 178481, 2796203  | 8971, 332207361361  | 47, 139, 3221, 7505944891  | 47, 3083, 31479823396757  | 47, 178481, 10052678938039  | 47, 1001523179, 23535794707  | 11111111111111111111111 | 829, 28878847, 3740221981231  | 47, 39891250417, 321218438243  | 1381, 2519545342349331183143  | 47, 461, 2347, 10627, 2249861, 14525237  | 829, 31741, 3046462151831565769  | 47, 277, 1013, 1657, 30269, 178481, 2796203  | 47, 26552618219228090162977481  | 47, 599, 7468009, 20801237997245359  | 277, 2347, 16497763013, 1335495402823  | 691, 1381, 46266279097921483078651  | 47, 19597, 139870566115103282847737  | 4463, 1323064018651, 60575166785239  | 461, 1289, 831603031789, 1920647391913  | 47, 124799, 304751, 58769065453824529
24  | 241 | 6481 | 97, 673  | 390001 | 1678321 | 73, 193, 409  | 433, 38737  | 97, 577, 769  | 99990001 | 10657, 20113  | 193, 2227777  | 815702161 | 1475750641 | 2562840001 | 193, 22253377  | 73, 1321, 72337  | 11019855601 | 4297, 3952393  | 31177, 821113  | 73, 518118697  | 191353, 286777  | 937, 83575993  | 97, 1134793633

## Decimal unique primes

At present, more than fifty unique primes or probable primes are known. However, there are only twenty-three unique primes below 10100. The following table lists all 23 unique primes below 10100 (sequence A040017 (sorted) and A007615 (ordered by period length) in OEIS) and their periods (sequence A051627 (ordered by corresponding primes) and A007498 (sorted) in OEIS)

Script error: No such module "Vertical header". | Prime

|

1 | 3
2 | 11
3 | 37
4 | 101
10 | 9,091
12 | 9,901
9 | 333,667
14 | 909,091
24 | 99,990,001
36 | 999,999,000,001
48 | 9,999,999,900,000,001
38 | 909,090,909,090,909,091
19 | 1,111,111,111,111,111,111
23 | 11,111,111,111,111,111,111,111
39 | 900,900,900,900,990,990,990,991
62 | 909,090,909,090,909,090,909,090,909,091
120 | 100,009,999,999,899,989,999,000,000,010,001
150 | 10,000,099,999,999,989,999,899,999,000,000,000,100,001
106 | 9,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,091
93 | 900,900,900,900,900,900,900,900,900,900,990,990,990,990,990,990,990,990,990,991
134 | 909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,091
294 | 142,857,157,142,857,142,856,999,999,985,714,285,714,285,857,142,857,142,855,714,285,571,428,571,428,572,857,143
196 | 999,999,999,999,990,000,000,000,000,099,999,999,999,999,000,000,000,000,009,999,999,999,999,900,000,000,000,001

The prime with period length 294 is similar[clarification needed] to the reciprocal of 7 (0.142857142857142857...)

Just after the table, the twenty-fourth unique prime has 128 digits and period length 320. It can be written as (932032)2 \+ 1, where a subscript number n indicates n consecutive copies of the digit or group of digits before the subscript.

Though they are rare, based on the occurrence of repunit primes and probable primes, it is conjectured strongly that there are infinitely many unique primes. (Any repunit prime is unique.)

(As of 2010) the repunit (10270343 – 1)/9 is the largest known probable unique prime.[2]

In 1996 the largest proven unique prime was (101132 + 1)/10001 or, using the notation above, (99990000)141 + 1\. It has 1128 digits. The record has been improved many times since then. (As of 2020) the largest proven unique prime is [math]\displaystyle{ \Phi_{39855}(-10) }[/math], it has 20160 digits.[3]

## Binary unique primes

The first unique primes in binary (base 2) are:

3, 5, 7, 11, 13, 17, 19, 31, 41, 43, 73, 127, 151, 241, 257, 331, 337, 683, ... (sequence A144755 (sorted) and A161509 (ordered by period length) in OEIS)

The period length of them are:

2, 4, 3, 10, 12, 8, 18, 5, 20, 14, 9, 7, 15, 24, 16, 30, 21, 22, ... (sequence A247071 (ordered by corresponding primes) and A161508 (sorted) in OEIS)

They include Fermat primes (the period length is a power of 2), Mersenne primes (the period length is a prime) and Wagstaff primes (the period length is twice an odd prime).

Additionally, if n is a natural number which is not equal to 1 or 6, than at least one prime have period n in base 2, because of the Zsigmondy theorem. Besides, if n is congruent to 4 (mod 8) and n > 20, then at least two primes have period n in base 2, (Thus, n is not a unique period in base 2) because of the Aurifeuillean factorization, for example, 113 ([math]\displaystyle{ =\Phi_{28L}(2) }[/math]) and 29 ([math]\displaystyle{ =\Phi_{28M}(2) }[/math]) both have period 28 in base 2, 37 ([math]\displaystyle{ =\Phi_{36L}(2) }[/math]) and 109 ([math]\displaystyle{ =\Phi_{36M}(2) }[/math]) both have period 36 in base 2, and that 397 ([math]\displaystyle{ =\Phi_{44L}(2) }[/math]) and 2113 ([math]\displaystyle{ =\Phi_{44M}(2) }[/math]) both have period 44 in base 2,

As shown above, a prime p is a unique prime of period n in base 2 if and only if there exists a natural number c such that

[math]\displaystyle{ \frac{\Phi_n(2)}{\gcd(\Phi_n(2),n)} = p^c. }[/math]

Empirically, it appears that most cases correspond to the denominator [math]\displaystyle{ \gcd(\Phi_n(2),n) }[/math] being 1, so we have a prime (or prime power) of form [math]\displaystyle{ \Phi_n(2) }[/math]. However, for n equal to 18, 20, 21, 54, 147, ... (sequence A333973 in the OEIS), we get a binary unique prime even though [math]\displaystyle{ \Phi_n(2) }[/math] has a small divisor (which is shared with n).

In fact, no prime with c > 1 (that is [math]\displaystyle{ \Phi_n(2)/\gcd(\Phi_n(2),n) }[/math] is a true power of p) have been discovered, and all known unique primes p have c = 1. It is conjectured that all unique primes have c = 1 (that is, all base-2 unique primes are not Wieferich primes).

As of September 2019, the largest known base 2 unique prime is 282589933-1, it is also the largest known prime. With an exception of Mersenne primes, the largest known probable base 2 unique prime is [math]\displaystyle{ \frac{2^{13372531}+1}{3} }[/math],[4] and the largest proven base 2 unique prime is [math]\displaystyle{ \frac{2^{83339}+1}{3} }[/math]. Besides, the largest known probable base 2 unique prime which is not Mersenne prime or Wagstaff prime is [math]\displaystyle{ \frac{2^{4101572}+1}{17} }[/math].

Similar to base 10, though they are rare (but more than the case to base 10), it is conjectured that there are infinitely many base 2 unique primes, because all Mersenne primes are unique in base 2, and it is conjectured they there are infinitely many Mersenne primes.[citation needed]

They divide none of overpseudoprimes to base 2, but every other odd prime number divide one overpseudoprime to base 2, because if and only if a composite number can be written as [math]\displaystyle{ \frac{\Phi_n(2)}{\gcd(\Phi_n(2),n)} }[/math], it is an overpseudoprime to base 2.[clarification needed]

There are 52 unique primes in base 2 below 264, they are:

Script error: No such module "Vertical header". | Prime (written in decimal) | Prime (written in binary)

|
|

2 | 3 | 11
4 | 5 | 101
3 | 7 | 111
10 | 11 | 1011
12 | 13 | 1101
8 | 17 | 1 0001
18 | 19 | 1 0011
5 | 31 | 1 1111
20 | 41 | 10 1001
14 | 43 | 10 1011
9 | 73 | 100 1001
7 | 127 | 111 1111
15 | 151 | 1001 0111
24 | 241 | 1111 0001
16 | 257 | 1 0000 0001
30 | 331 | 1 0100 1011
21 | 337 | 1 0101 0001
22 | 683 | 10 1010 1011
26 | 2,731 | 1010 1010 1011
42 | 5,419 | 1 0101 0010 1011
13 | 8,191 | 1 1111 1111 1111
34 | 43,691 | 1010 1010 1010 1011
40 | 61,681 | 1111 0000 1111 0001
32 | 65,537 | 1 0000 0000 0000 0001
54 | 87,211 | 1 0101 0100 1010 1011
17 | 131,071 | 1 1111 1111 1111 1111
38 | 174,763 | 10 1010 1010 1010 1011
27 | 262,657 | 100 0000 0010 0000 0001
19 | 524,287 | 111 1111 1111 1111 1111
33 | 599,479 | 1001 0010 0101 1011 0111
46 | 2,796,203 | 10 1010 1010 1010 1010 1011
56 | 15,790,321 | 1111 0000 1111 0000 1111 0001
90 | 18,837,001 | 1 0001 1111 0110 1110 0000 1001
78 | 22,366,891 | 1 0101 0101 0100 1010 1010 1011
62 | 715,827,883 | 10 1010 1010 1010 1010 1010 1010 1011
31 | 2,147,483,647 | 111 1111 1111 1111 1111 1111 1111 1111
80 | 4,278,255,361 | 1111 1111 0000 0000 1111 1111 0000 0001
120 | 4,562,284,561 | 1 0000 1111 1110 1110 1111 0000 0001 0001
126 | 77,158,673,929 | 1 0001 1111 0111 0000 0011 1110 1110 0000 1001
150 | 1,133,836,730,401 | 1 0000 0111 1111 1101 1110 1111 1000 0000 0010 0001
86 | 2,932,031,007,403 | 10 1010 1010 1010 1010 1010 1010 1010 1010 1010 1011
98 | 4,363,953,127,297 | 11 1111 1000 0000 1111 1110 0000 0011 1111 1000 0001
49 | 4,432,676,798,593 | 100 0000 1000 0001 0000 0010 0000 0100 0000 1000 0001
69 | 10,052,678,938,039 | 1001 0010 0100 1001 0010 0101 1011 0110 1101 1011 0111
65 | 145,295,143,558,111 | 1000 0100 0010 0101 0010 1001 0110 1011 0101 1011 1101 1111
174 | 96,076,791,871,613,611 | 1 0101 0101 0101 0101 0101 0101 0100 1010 1010 1010 1010 1010 1010 1011
77 | 581,283,643,249,112,959 | 1000 0001 0001 0010 0010 0110 0100 1100 1101 1001 1011 1011 0111 0111 1111
93 | 658,812,288,653,553,079 | 1001 0010 0100 1001 0010 0100 1001 0011 0110 1101 1011 0110 1101 1011 0111
122 | 768,614,336,404,564,651 | 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1011
61 | 2,305,843,009,213,693,951 | 1 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111
85 | 9,520,972,806,333,758,431 | 1000 0100 0010 0001 0100 1010 0101 0010 1011 0101 1010 1101 0111 1011 1101 1111
192 | 18,446,744,069,414,584,321 | 1111 1111 1111 1111 1111 1111 1111 1111 0000 0000 0000 0000 0000 0000 0000 0001

After the table, the next 10 binary unique prime have period length 170, 234, 158, 165, 147, 129, 184, 89, 208, and 312. Besides, the bits (digits in binary) of them are 65, 73, 78, 81, 82, 84, 88, 89, 96, and 97.

## Bi-unique primes

Bi-unique primes are pairs of primes that share a period length with each other, but with no other primes. For example, in binary, the bi-unique primes with at least one prime less than 10000 are:

prime
p | the only other prime having the same period as p | period
length

|
|

23  | 89  | 11
29  | 113  | 28
37  | 109  | 36
47  | 178481  | 23
59  | 3033169  | 58
61  | 1321  | 60
67  | 20857  | 66
71  | 122921  | 35
79  | 121369  | 39
83  | 8831418697  | 82
89  | 23  | 11
97  | 673  | 48
107  | 28059810762433  | 106
109  | 37  | 36
113  | 29  | 28
139  | 168749965921  | 138
167  | 57912614113275649087721  | 83
193  | 22253377  | 96
223  | 616318177  | 37
251  | 4051  | 50
263  | 10350794431055162386718619237468234569  | 131
281  | 86171  | 70
283  | 165768537521  | 94
353  | 2931542417  | 88
397  | 2113  | 44
433  | 38737  | 72
463  | 4982397651178256151338302204762057  | 231
571  | 160465489  | 114
577  | 487824887233  | 144
601  | 1801  | 25
607  | 1512768222413735255864403005264105839324374778520631853993  | 303
631  | 23311  | 45
641  | 6700417  | 64
643  | 84115747449047881488635567801  | 214
673  | 97  | 48
727  | 1786393878363164227858270210279  | 121
751  | 2139731020464054092520609592459940706818275139793055476751  | 375
769  | 442499826945303593556473164314770689  | 384
919  | 75582488424179347083438319  | 153
1039  | 19709014643115560219397264671577125505264032974428376489237001990435774189483906244488746953221813209  | 519
1291  | 83861817871925183739792206470703862766563053456867813459969184678546547694793573468589875745315081  | 1290
1321  | 61  | 60
1327  | 2365454398418399772605086209214363458552839866247069233  | 221
1429  | 14449  | 84
1471  | 252359902034571016856214298851708529738525821631  | 245
1543  | 4965395030068548134274243124972075225434447114375481299036593442726326832727934403424309955102162841656341524725641213163998408700663382552888660520657  | 771
1697  | 99335205800663868215396640964567095667094665346141013294320587365443384719802857319737050495099341955640963272958071602273  | 848
1753  | 1795918038741070627  | 146
1777  | 25781083  | 74
1801  | 601  | 25
2113  | 397  | 44
2281  | 3011347479614249131  | 190
2801  | 1114513219367157067542813609361306957257890531134775327875067038594481393220804051366788787128409731513666376851495151281817670381468528387601  | 1400
2971  | 48912491  | 110
3011  | 631215008947706187342830494125660733360092019659681922883823392015121754384870744044074337887482936870852519582960673945561810148710850934449712549090934572292098088972061029650939105592263256293676274598529593937386833315889748213948490958132757432166701901197169972066727635929332437543971934775961  | 3010
3259  | 960843850986532976532466235773483492840618819232206145010143480044702708779967241439519037158800917230289  | 1086
3361  | 88959882481  | 168
3967  | 3296810823331827444014404831943558588631803435050404237042485765714486337505843011741487225539321479275976317423474114853376321380782906502106758766783934866952124117240484839332668914566806988602931402117416523955329423560856334826333176954575294550104263404414368761262079586842542586869780254842277261781328657636993064897732127711363870426953852536828242291991249685206783121190349820804553  | 1983
4051  | 251  | 50
4129  | 33770734168253651800370989375796994825389296318018601048482005531172856260013942500368975908606689  | 688
4177  | 9857737155463  | 87
4523  | 106788290443848295284382097033  | 266
4561  | 51049903050598156013062477654241640657829025002976204451060261008689478158715729745160924860467530309657376827104233308157772350164622158651187694109112727796663977157921  | 2280
4871  | 82033219963138371097689272308258116841679442057301643873942124991182012434598644913857356023840478815121709542915222280972560231358838127531337  | 487
5153  | 54410972897  | 112
5281  | 860573414369008969457638101533827364704684164286188824383996871471626764468219429432850649798234488791036977295952185305281368586922360240161830570497885830241813162641447900350702795124321  | 2640
5347  | 242099935645987  | 198
5431  | 9496792988973395279834809661569251320544014305629535339041176003993804676485199470146061766714674245857484692035146769896735263094609746643655454990307740324793870789211183639302657602006206727998357155351588374904926882678074633252877772442134938712719216422825919305415646969838444956229559805502059906228691551284152908049964563516405524127028294884716508273187514617905113012642316992618568629664046915514871575875088457784721  | 2715
5881  | 23618256244840618857212522155851714598259422753496906641681177748710460515038403366198473773770441  | 1470
6043  | 4475130366518102084427698737  | 318
6659  | 67348091890626757137914773048080151982788009808953349522971703676209072447919253736713943719839321930921001920443146832964494535806286153336758808764827052922284527987735369082658226155752758837734585788583920437031679967832798697874531506375848295306002069974292647012532975382657869395896549536084026643086849168707638035984652951756449232381922841508279043449553683642165895518852273616853752192626547082225232322662203349617421624450106130361133033050996977517456780759336980017504035553443204836636663583950658161718264375035034007950563418777684540446570742277682688819816930529249402141227674467861784042184664703273065772145626307008333021727910295689307897812592176340797189662547498298629041419687123412984210580325376481846316396566413701109848755348878790562296275295266190788880122151835567771665815565611863261470167857288685014242115505182159685753576612239477286620238583071292970734389580521730540789853959607322402465845627773409459421340250476125665859926003121138412497735360569  | 6658
6719  | 215006106257113223254503015023149432126193150293791416185445173578281597218315377296589584591228602041183907532584815068471747291177386898925622477208530115714962355294842135137890474394949339249259335407710018584480055157825387089416912233252714054247018216597994795059161567922302450277281351135838393171424038832688432240078361264161523904355539085927738753968157018550258476163852090826756157915705283413226000816151712543838581066281600650278690534719371112997393190721068136840596790525950480851370510277560248341182341805553054000587378384785994695875587394905226703149605689830257768229987714770949192483302583569799141079867597051190134078718011730508482542567284418838119000563443985593691221203060137047648713095502877775290062907208508727269017130916691676838817452529964938878349395785642430571852241837461604136374448443730175081889502056290512497717177577492736555784081731998565765598518104822516520340301701034123926767472784665776779480628821628279687651736198541330802238405154786248043073  | 3359
7487  | 26828803997912886929710867041891989490486893845712448833  | 197
8929  | 197107422273014301919781414466039325387889623676342705850752210599969  | 496
8969  | 10508537584872980049787749414505440238543661684506416445249892188329191267897669657242625405655025902294996965713681247700894953567276596965114308183649957469931262029470372188492494505614207827774171575432114297123003373257035070542940532411186322417809411123684246738342720455933424175399671044286557638075591  | 1121
9547  | 1621441292160739312484402643488810210953460916758334047593952342310982348899125523375207637304333778211869062392988099059802528019593682234941755422758885656395068722385037980657466257618112582188770921312100125511337836412531718154395821529210922389443733616354268820219577863577759459082447218927273695668223251258943006743614909639761127161704816862626236353032622115795192245125083091261029053988053316433377173895018793740052548266015018756763150731725385456332232982433576015547722563978072554378000015707071821371450842910648052930276764535303424167478747579771592484270800978561959411183367133498969236434846108865206764889977963554070295936092795484663326925724277620386077381551473009733178990983013487085676185144378484902884955972104873606558173086267499547566081780818064857671567480196001693236835136811036110768546793929610732909274227296407079545788520811837495181586420117807667033593394473  | 9546

Although there are 1228 odd primes below 10000, only 21 of them are unique and 76 of them are bi-unique in binary.

A classic example of binary bi-unique primes are

46817226351072265620777670675006972301618979214252832875068976303839400413682313921168154465151768472420980044715745858522803980473207943564433 (143 digits)

and

527739642811233917558838216073534609312522896254707972010583175760467054896492872702786549764052643493511382273226052631979775533936351462037464331880467187717179256707148303247 (177 digits)

they are the two prime factors of the Mersenne number 21061−1. [5] Thus, the period
length of them is 1061.

As of October 2016, the largest known probable binary bi-unique prime is [math]\displaystyle{ \frac{2^{5240707}-1}{75392810903} }[/math], [6] it has a period
length of 5240707 shares with only the prime 75392810903.

Similarly, we can define "tri-unique primes" as a triple of primes having a period
length shared by no other primes. In binary, the first few tri-unique primes are:

prime p | the only two other primes having the same period as p | period
length

|
|

53  | 157, 1613  | 52
101  | 8101, 268501  | 100
103  | 2143, 11119  | 51
131  | 409891, 7623851  | 130
137  | 953, 26317  | 68
157  | 53, 1613  | 52
163  | 135433, 272010961  | 162
179  | 62020897, 18584774046020617  | 178
181  | 54001, 29247661  | 180
191  | 420778751, 30327152671  | 95
197  | 19707683773, 4981857697937  | 196
199  | 153649, 33057806959  | 99
211  | 664441, 1564921  | 210
229  | 457, 525313  | 76
233  | 1103, 2089  | 29
271  | 348031, 49971617830801  | 135
307  | 2857, 6529  | 102
317  | 381364611866507317969, 604462909806215075725313  | 316
359  | 1433, 1489459109360039866456940197095433721664951999121  | 179
367  | 55633, 37201708625305146303973352041  | 183
373  | 951088215727633, 4611545283086450689  | 372
419  | 3410623284654639440707, 1607792018780394024095514317003  | 418
421  | 146919792181, 1041815865690181  | 420
431  | 9719, 2099863  | 43
439  | 2298041, 9361973132609  | 73
443  | 4714692062809, 4507513575406446515845401458366741487526913  | 442
457  | 229, 525313  | 76
467  | 27961, 352369374013660139472574531568890678155040563007620742839120913  | 466
491  | 15162868758218274451, 50647282035796125885000330641  | 490

In binary, the smallest n-unique prime are

3, 23, 53, 149, 269, 461, 619, 389, ...

In binary, the period length of odd primes are: (sequence A014664 in the OEIS)

prime  | period
length  | prime  | period
length  | prime  | period
length  | prime  | period
length  | prime  | period
length  | prime  | period
length  | prime  | period
length

|
|
|
|
|
|
|
|
|
|
|
|
|

3  | 2  | 79  | 39  | 181  | 180  | 293  | 292  | 421  | 420  | 557  | 556  | 673  | 48
5  | 4  | 83  | 82  | 191  | 95  | 307  | 102  | 431  | 43  | 563  | 562  | 677  | 676
7  | 3  | 89  | 11  | 193  | 96  | 311  | 155  | 433  | 72  | 569  | 284  | 683  | 22
11  | 10  | 97  | 48  | 197  | 196  | 313  | 156  | 439  | 73  | 571  | 114  | 691  | 230
13  | 12  | 101  | 100  | 199  | 99  | 317  | 316  | 443  | 442  | 577  | 144  | 701  | 700
17  | 8  | 103  | 51  | 211  | 210  | 331  | 30  | 449  | 224  | 587  | 586  | 709  | 708
19  | 18  | 107  | 106  | 223  | 37  | 337  | 21  | 457  | 76  | 593  | 148  | 719  | 359
23  | 11  | 109  | 36  | 227  | 226  | 347  | 346  | 461  | 460  | 599  | 299  | 727  | 121
29  | 28  | 113  | 28  | 229  | 76  | 349  | 348  | 463  | 231  | 601  | 25  | 733  | 244
31  | 5  | 127  | 7  | 233  | 29  | 353  | 88  | 467  | 466  | 607  | 303  | 739  | 246
37  | 36  | 131  | 130  | 239  | 119  | 359  | 179  | 479  | 239  | 613  | 612  | 743  | 371
41  | 20  | 137  | 68  | 241  | 24  | 367  | 183  | 487  | 243  | 617  | 154  | 751  | 375
43  | 14  | 139  | 138  | 251  | 50  | 373  | 372  | 491  | 490  | 619  | 618  | 757  | 756
47  | 23  | 149  | 148  | 257  | 16  | 379  | 378  | 499  | 166  | 631  | 45  | 761  | 380
53  | 52  | 151  | 15  | 263  | 131  | 383  | 191  | 503  | 251  | 641  | 64  | 769  | 384
59  | 58  | 157  | 52  | 269  | 268  | 389  | 388  | 509  | 508  | 643  | 214  | 773  | 772
61  | 60  | 163  | 162  | 271  | 135  | 397  | 44  | 521  | 260  | 647  | 323  | 787  | 786
67  | 66  | 167  | 83  | 277  | 92  | 401  | 200  | 523  | 522  | 653  | 652  | 797  | 796
71  | 35  | 173  | 172  | 281  | 70  | 409  | 204  | 541  | 540  | 659  | 658  | 809  | 404
73  | 9  | 179  | 178  | 283  | 94  | 419  | 418  | 547  | 546  | 661  | 660  | 811  | 270

In binary, the primes with given period length are: (sequence A108974 in the OEIS)

period
length  | prime(s)  | period
length  | prime(s)  | period
length  | prime(s)  | period
length  | prime(s)

|
|
|
|
|
|
|

1  | (none)  | 26  | 2731  | 51  | 103, 2143, 11119  | 76  | 229, 457, 525313
2  | 3  | 27  | 262657  | 52  | 53, 157, 1613  | 77  | 581283643249112959
3  | 7  | 28  | 29, 113  | 53  | 6361, 69431, 20394401  | 78  | 22366891
4  | 5  | 29  | 233, 1103, 2089  | 54  | 87211  | 79  | 2687, 202029703, 1113491139767
5  | 31  | 30  | 331  | 55  | 881, 3191, 201961  | 80  | 4278255361
6  | (none)  | 31  | 2147483647  | 56  | 15790321  | 81  | 2593, 71119, 97685839
7  | 127  | 32  | 65537  | 57  | 32377, 1212847  | 82  | 83, 8831418697
8  | 17  | 33  | 599479  | 58  | 59, 3033169  | 83  | 167, 57912614113275649087721
9  | 73  | 34  | 43691  | 59  | 179951, 3203431780337  | 84  | 1429, 14449
10  | 11  | 35  | 71, 122921  | 60  | 61, 1321  | 85  | 9520972806333758431
11  | 23, 89  | 36  | 37, 109  | 61  | 2305843009213693951  | 86  | 2932031007403
12  | 13  | 37  | 223, 616318177  | 62  | 715827883  | 87  | 4177, 9857737155463
13  | 8191  | 38  | 174763  | 63  | 92737, 649657  | 88  | 353, 2931542417
14  | 43  | 39  | 79, 121369  | 64  | 641, 6700417  | 89  | 618970019642690137449562111
15  | 151  | 40  | 61681  | 65  | 145295143558111  | 90  | 18837001
16  | 257  | 41  | 13367, 164511353  | 66  | 67, 20857  | 91  | 911, 112901153, 23140471537
17  | 131071  | 42  | 5419  | 67  | 193707721, 761838257287  | 92  | 277, 1013, 1657, 30269
18  | 19  | 43  | 431, 9719, 2099863  | 68  | 137, 953, 26317  | 93  | 658812288653553079
19  | 524287  | 44  | 397, 2113  | 69  | 10052678938039  | 94  | 283, 165768537521
20  | 41  | 45  | 631, 23311  | 70  | 281, 86171  | 95  | 191, 420778751, 30327152671
21  | 337  | 46  | 2796203  | 71  | 228479, 48544121, 212885833  | 96  | 193, 22253377
22  | 683  | 47  | 2351, 4513, 13264529  | 72  | 433, 38737  | 97  | 11447, 13842607235828485645766393
23  | 47, 178481  | 48  | 97, 673  | 73  | 439, 2298041, 9361973132609  | 98  | 4363953127297
24  | 241  | 49  | 4432676798593  | 74  | 1777, 25781083  | 99  | 199, 153649, 33057806959
25  | 601, 1801  | 50  | 251, 4051  | 75  | 100801, 10567201  | 100  | 101, 8101, 268501

## Period lengths

Table of period lengths from 1 to 100 (unique primes are bold)

| Period

length

| Primes  | Period

length

| Primes  | Period

length

| Primes  | Period

length

| Primes  | Period

length

| Primes

|
|
|
|
|
|
|
|
|

1  | 3 | 21  | 43, 1933, 10838689  | 41  | 83, 1231, 538987, 201763709900322803748657942361  | 61  | 733, 4637, 329401, 974293, 1360682471, 106007173861643, 7061709990156159479  | 81  | 163, 9397, 2462401, 676421558270641, 130654897808007778425046117
2  | 11 | 22  | 23, 4093, 8779  | 42  | 127, 2689, 459691  | 62  | 909090909090909090909090909091 | 82  | 2670502781396266997, 3404193829806058997303
3  | 37 | 23  | 11111111111111111111111 | 43  | 173, 1527791, 1963506722254397, 2140992015395526641  | 63  | 10837, 23311, 45613, 45121231, 1921436048294281  | 83  | 3367147378267, 9512538508624154373682136329, 346895716385857804544741137394505425384477
4  | 101 | 24  | 99990001 | 44  | 89, 1052788969, 1056689261  | 64  | 19841, 976193, 6187457, 834427406578561  | 84  | 226549, 4458192223320340849
5  | 41, 271  | 25  | 21401, 25601, 182521213001  | 45  | 238681, 4185502830133110721  | 65  | 162503518711, 5538396997364024056286510640780600481  | 85  | 262533041, 8119594779271, 4222100119405530170179331190291488789678081
6  | 7, 13  | 26  | 859, 1058313049  | 46  | 47, 139, 2531, 549797184491917  | 66  | 599144041, 183411838171  | 86  | 57009401, 2182600451, 7306116556571817748755241
7  | 239, 4649  | 27  | 757, 440334654777631  | 47  | 35121409, 316362908763458525001406154038726382279  | 67  | 493121, 79863595778924342083, 28213380943176667001263153660999177245677  | 87  | 4003, 72559, 310170251658029759045157793237339498342763245483
8  | 73, 137  | 28  | 29, 281, 121499449  | 48  | 9999999900000001 | 68  | 28559389, 1491383821, 2324557465671829  | 88  | 617, 16205834846012967584927082656402106953
9  | 333667 | 29  | 3191, 16763, 43037, 62003, 77843839397  | 49  | 505885997, 1976730144598190963568023014679333  | 69  | 277, 203864078068831, 1595352086329224644348978893  | 89  | 497867, 103733951, 104984505733, 5078554966026315671444089, 403513310222809053284932818475878953159
10  | 9091 | 30  | 211, 241, 2161  | 50  | 251, 5051, 78875943472201  | 70  | 4147571, 265212793249617641  | 90  | 29611, 3762091, 8985695684401
11  | 21649, 513239  | 31  | 2791, 6943319, 57336415063790604359  | 51  | 613, 210631, 52986961, 13168164561429877  | 71  | 241573142393627673576957439049, 45994811347886846310221728895223034301839  | 91  | 547, 14197, 17837, 4262077, 43442141653, 316877365766624209, 110742186470530054291318013
12  | 9901 | 32  | 353, 449, 641, 1409, 69857  | 52  | 521, 1900381976777332243781  | 72  | 3169, 98641, 3199044596370769  | 92  | 1289, 18371524594609, 4181003300071669867932658901
13  | 53, 79, 265371653  | 33  | 67, 1344628210313298373  | 53  | 107, 1659431, 1325815267337711173, 47198858799491425660200071  | 73  | 12171337159, 1855193842151350117, 49207341634646326934001739482502131487446637  | 93  | 900900900900900900900900900900990990990990990990990990990991
14  | 909091 | 34  | 103, 4013, 21993833369  | 54  | 70541929, 14175966169  | 74  | 7253, 422650073734453, 296557347313446299  | 94  | 6299, 4855067598095567, 297262705009139006771611927
15  | 31, 2906161  | 35  | 71, 123551, 102598800232111471  | 55  | 1321, 62921, 83251631, 1300635692678058358830121  | 75  | 151, 4201, 15763985553739191709164170940063151  | 95  | 191, 59281, 63841, 1289981231950849543985493631, 965194617121640791456070347951751
16  | 17, 5882353  | 36  | 999999000001 | 56  | 7841, 127522001020150503761  | 76  | 722817036322379041, 1369778187490592461  | 96  | 97, 206209, 66554101249, 75118313082913
17  | 2071723, 5363222357  | 37  | 2028119, 247629013, 2212394296770203368013  | 57  | 21319, 10749631, 3931123022305129377976519  | 77  | 5237, 42043, 29920507, 136614668576002329371496447555915740910181043  | 97  | 12004721, 92556179448994367391887834053878562534782033760810527051075248738484727059555245899601591
18  | 19, 52579  | 38  | 909090909090909091 | 58  | 59, 154083204930662557781201849  | 78  | 157, 6397, 216451, 388847808493  | 98  | 197, 5076141624365532994918781726395939035533
19  | 1111111111111111111 | 39  | 900900900900990990990991 | 59  | 2559647034361, 4340876285657460212144534289928559826755746751  | 79  | 317, 6163, 10271, 307627, 49172195536083790769, 3660574762725521461527140564875080461079917  | 99  | 199, 397, 34849, 362853724342990469324766235474268869786311886053883
20  | 3541, 27961  | 40  | 1676321, 5964848081  | 60  | 61, 4188901, 39526741  | 80  | 5070721, 19721061166646717498359681  | 100  | 60101, 7019801, 14103673319201, 1680588011350901

## Unique prime in various bases

base  | unique period length (searched up to 4096)

|

2 | 2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 27, 30, 31, 32, 33, 34, 38, 40, 42, 46, 49, 54, 56, 61, 62, 65, 69, 77, 78, 80, 85, 86, 89, 90, 93, 98, 107, 120, 122, 126, 127, 129, 133, 145, 147, 150, 158, 165, 170, 174, 184, 192, 195, 202, 208, 234, 254, 261, 280, 296, 312, 322, 334, 342, 345, 366, 374, 382, 398, 410, 414, 425, 447, 471, 507, 521, 550, 567, 579, 590, 600, 602, 607, 626, 690, 694, 712, 745, 795, 816, 889, 897, 909, 954, 990, 1106, 1192, 1224, 1230, 1279, 1384, 1386, 1402, 1464, 1512, 1554, 1562, 1600, 1670, 1683, 1727, 1781, 1834, 1904, 1990, 1992, 2008, 2037, 2203, 2281, 2298, 2353, 2406, 2456, 2499, 2536, 2838, 3006, 3074, 3217, 3415, 3418, 3481, 3766, 3817, 3927, ...
3 | 1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 20, 21, 24, 26, 32, 33, 36, 40, 46, 60, 63, 64, 70, 71, 72, 86, 103, 108, 128, 130, 132, 143, 145, 154, 161, 236, 255, 261, 276, 279, 287, 304, 364, 430, 464, 513, 528, 541, 562, 665, 672, 680, 707, 718, 747, 760, 782, 828, 875, 892, 974, 984, 987, 1037, 1058, 1070, 1073, 1080, 1091, 1154, 1248, 1367, 1426, 1440, 1462, 1524, 1598, 1623, 1627, 1863, 1985, 2132, 2188, 2196, 2340, 2460, 2508, 2626, 2640, 2739, 2856, 3092, 3158, 3262, 3315, 3326, 3482, 3638, 3982, 4018, 4036, ...
4 | 1, 2, 3, 4, 6, 8, 10, 12, 16, 20, 28, 40, 60, 92, 96, 104, 140, 148, 156, 300, 356, 408, 596, 612, 692, 732, 756, 800, 952, 996, 1004, 1228, 1268, 2240, 2532, 3060, 3796, 3824, 3944, ...
5 | 1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 13, 18, 24, 28, 47, 48, 49, 56, 57, 88, 90, 92, 108, 110, 116, 120, 127, 134, 141, 149, 161, 171, 181, 198, 202, 206, 236, 248, 288, 357, 384, 420, 458, 500, 530, 536, 619, 620, 694, 798, 897, 929, 981, 992, 1064, 1134, 1230, 1670, 1807, 2094, 2162, 2369, 2516, 2649, 2988, 3407, 3888, ...
6 | 1, 2, 3, 4, 5, 6, 7, 8, 18, 21, 22, 24, 29, 30, 42, 50, 62, 71, 86, 90, 94, 118, 124, 127, 129, 144, 154, 186, 192, 214, 271, 354, 360, 411, 480, 509, 558, 575, 663, 764, 814, 825, 874, 1028, 1049, 1050, 1102, 1113, 1131, 1158, 1376, 1464, 1468, 1535, 1622, 1782, 1834, 1924, 2096, 2176, 2409, 2464, 2816, 3013, 3438, 3453, 3663, ...
7 | 3, 5, 6, 8, 13, 18, 21, 28, 30, 34, 36, 46, 48, 50, 54, 55, 58, 63, 76, 84, 94, 105, 122, 131, 148, 149, 224, 280, 288, 296, 332, 352, 456, 528, 531, 581, 650, 654, 730, 740, 759, 1026, 1047, 1065, 1460, 1660, 1699, 1959, 2067, 2260, 2380, 2665, 2890, 3238, 4020, ...
8 | 1, 2, 3, 6, 9, 18, 30, 42, 78, 87, 114, 138, 189, 303, 318, 330, 408, 462, 504, 561, 1002, 1389, 1746, 1794, 2040, 2418, 2790, 3894, 4077, ...
9 | 1, 2, 4, 6, 10, 12, 16, 18, 20, 30, 32, 36, 54, 64, 66, 118, 138, 152, 182, 232, 264, 336, 340, 380, 414, 446, 492, 540, 624, 720, 762, 1066, 1094, 1098, 1170, 1230, 1254, 1320, 1428, 1546, 2018, 2574, 2724, 2804, 2920, 3074, 3316, 3646, ...
10 | 1, 2, 3, 4, 9, 10, 12, 14, 19, 23, 24, 36, 38, 39, 48, 62, 93, 106, 120, 134, 150, 196, 294, 317, 320, 385, 586, 597, 654, 738, 945, 1031, 1172, 1282, 1404, 1426, 1452, 1521, 1752, 1812, 1836, 1844, 1862, 2134, 2232, 2264, 2667, 3750, 3903, 3927, ...
11 | 2, 4, 5, 6, 8, 9, 10, 14, 15, 17, 18, 19, 20, 27, 36, 42, 45, 52, 60, 73, 91, 104, 139, 205, 234, 246, 318, 358, 388, 403, 458, 552, 810, 855, 878, 907, 1114, 1131, 1220, 1272, 1431, 1470, 1568, 1614, 1688, 1696, 1907, 2029, 2136, 2288, 2535, 2577, ...
12 | 1, 2, 3, 5, 10, 12, 19, 20, 21, 22, 56, 60, 63, 70, 80, 84, 92, 97, 109, 111, 123, 164, 189, 218, 276, 317, 353, 364, 386, 405, 456, 511, 636, 675, 701, 793, 945, 1090, 1268, 1272, 1971, 2088, 2368, 2482, 2893, 2966, 3290, ...
13 | 2, 3, 5, 6, 7, 8, 9, 12, 16, 22, 24, 28, 33, 34, 38, 78, 80, 102, 137, 140, 147, 224, 230, 283, 304, 341, 360, 372, 384, 418, 420, 436, 483, 568, 570, 594, 737, 744, 855, 883, 991, 1021, 1193, 1222, 1615, 1628, 1838, 2032, 2146, 2302, 2530, 2830, 2958, 3030, 3528, 3671, 3885, ...
14 | 1, 3, 4, 6, 7, 14, 19, 24, 31, 33, 35, 36, 41, 55, 60, 106, 114, 129, 152, 153, 172, 222, 265, 286, 400, 448, 560, 584, 864, 1006, 1335, 1363, 1520, 1536, 1659, 1862, 1925, 2332, 2458, 2687, 3381, 3512, 3870, 3976, ...
15 | 3, 4, 6, 7, 14, 24, 43, 54, 58, 73, 85, 93, 102, 184, 220, 221, 228, 232, 247, 291, 305, 486, 487, 505, 551, 552, 590, 1029, 1194, 1274, 1406, 1444, 1532, 1548, 1748, 1986, 2093, 2182, 2202, 2579, 2781, 3054, 3239, 3696, ...
16 | 2, 4, 6, 8, 10, 14, 20, 30, 46, 48, 52, 70, 74, 78, 150, 178, 204, 298, 306, 346, 366, 378, 400, 476, 498, 502, 614, 634, 1120, 1266, 1530, 1898, 1912, 1972, 2548, 2770, 3738, 3850, ...
17 | 1, 2, 3, 5, 7, 8, 11, 12, 14, 15, 34, 42, 46, 47, 48, 50, 71, 77, 94, 110, 114, 147, 154, 176, 228, 235, 258, 275, 338, 350, 419, 450, 480, 515, 589, 624, 666, 716, 724, 810, 815, 1232, 1490, 1934, 2106, 2391, 2732, 2904, 3462, 3912, 4053, ...
18 | 1, 2, 3, 6, 14, 17, 21, 24, 30, 33, 38, 45, 46, 72, 78, 114, 146, 168, 288, 414, 440, 448, 665, 792, 801, 816, 975, 1165, 1176, 1267, 1466, 1513, 1882, 1920, 1998, 2194, 2272, 2643, 2800, 2946, 3434, 3504, 3813, 3866, 3957, ...
19 | 2, 3, 4, 6, 19, 20, 31, 34, 47, 56, 59, 61, 70, 74, 91, 92, 96, 98, 107, 120, 145, 156, 168, 242, 276, 314, 326, 337, 387, 565, 602, 612, 892, 984, 1061, 1067, 1079, 1262, 1328, 2356, 3033, 3419, 3501, 3963, ...
20 | 1, 3, 4, 6, 8, 9, 10, 11, 17, 30, 98, 100, 110, 126, 154, 158, 160, 168, 178, 182, 228, 266, 270, 280, 340, 416, 480, 574, 774, 980, 1052, 1139, 1338, 1418, 1474, 1487, 1594, 1902, 2326, 3112, 3520, 3808, 3830, ...
21 | 2, 3, 5, 6, 8, 9, 10, 11, 14, 17, 26, 43, 64, 74, 81, 104, 192, 271, 321, 335, 348, 404, 437, 445, 516, 671, 694, 788, 1788, 1943, 2343, 2742, 3031, 3135, ...
22 | 2, 5, 6, 7, 10, 21, 25, 26, 69, 79, 86, 93, 100, 101, 154, 158, 161, 171, 202, 214, 294, 354, 359, 424, 454, 602, 687, 706, 744, 857, 1028, 1074, 1136, 1150, 1345, 1408, 1525, 1572, 1578, 1988, 2142, 2665, ...
23 | 2, 5, 8, 11, 15, 22, 26, 39, 42, 45, 54, 56, 132, 134, 145, 147, 196, 212, 218, 252, 343, 580, 662, 816, 820, 846, 1078, 1092, 1174, 1189, 1548, 1632, 2040, 2180, 2348, 2732, 3100, 3181, 4010, ...
24 | 1, 2, 3, 4, 5, 8, 14, 19, 22, 38, 45, 53, 54, 70, 71, 117, 140, 144, 169, 186, 192, 195, 196, 430, 653, 661, 744, 834, 855, 870, 927, 1128, 1158, 1390, 1516, 1555, 1617, 1844, 2022, 2060, 2208, 2812, 3153, 3952, ...
25 | 2, 4, 6, 12, 14, 24, 28, 44, 46, 54, 58, 60, 118, 124, 144, 192, 210, 250, 268, 310, 496, 532, 1258, 1494, 1944, 2050, 2498, 2728, 3324, 3418, 3646, 3862, 4014, ...
26 | 1, 2, 4, 7, 9, 18, 20, 22, 24, 30, 43, 69, 132, 140, 186, 200, 210, 218, 267, 347, 454, 495, 554, 585, 645, 694, 980, 1028, 1060, 1098, 1432, 1714, 1828, 3513, 3786, ...
27 | 2, 3, 12, 21, 24, 36, 87, 93, 171, 249, 276, 360, 480, 621, 732, 780, 1716, 3843, ...
28 | 1, 2, 3, 5, 6, 8, 17, 21, 38, 81, 91, 96, 102, 132, 148, 156, 240, 258, 260, 276, 457, 464, 465, 500, 506, 535, 684, 746, 838, 930, 982, 1015, 1189, 1296, 1335, 1345, 1390, 1423, 2062, 2723, 2893, 3078, ...
29 | 4, 5, 6, 7, 8, 14, 30, 32, 39, 45, 50, 76, 116, 151, 222, 357, 402, 462, 570, 588, 636, 671, 695, 844, 1498, 1650, 1770, 3175, 3195, 3312, 3538, 3719, ...
30 | 1, 2, 5, 9, 11, 12, 21, 36, 51, 64, 91, 163, 174, 195, 230, 278, 318, 342, 346, 424, 530, 569, 578, 795, 984, 1094, 1167, 1335, 1564, 1605, 1658, 1789, 2159, 2204, 2225, 3366, 3458, 3615, ...
31 | 3, 7, 12, 17, 24, 30, 31, 33, 40, 176, 218, 308, 404, 420, 630, 693, 890, 915, 922, 1475, 2122, 2185, 2487, 2541, 2907, 3387, 4055, ...
32 | 1, 6, 30, 85, 110, 120, 320, 1050, 1065, 1385, 2490, 3080, 3920, ...
33 | 1, 2, 3, 10, 16, 25, 28, 30, 35, 36, 45, 56, 76, 87, 110, 134, 135, 197, 200, 220, 228, 314, 324, 330, 396, 498, 583, 624, 725, 806, 940, 1145, 1240, 1644, 1750, 2171, 2268, 2675, 2781, 2790, 2808, 3581, ...
34 | 3, 6, 8, 10, 13, 20, 24, 56, 87, 154, 164, 196, 282, 363, 428, 652, 744, 780, 860, 902, 952, 1178, 1493, 1540, 1643, 1904, 2184, 2277, 2468, 2943, ...
35 | 2, 4, 6, 8, 18, 21, 22, 26, 42, 128, 154, 158, 170, 180, 184, 192, 254, 313, 450, 624, 737, 762, 798, 874, 912, 1002, 1006, 1098, 1234, 1297, 1418, 1714, 1926, 2325, 2343, 2368, 2998, 3567, 4064, ...
36 | 2, 4, 12, 62, 72, 96, 180, 240, 382, 514, 688, 732, 734, 962, 1048, 1088, 1232, 1408, 2088, 2176, 2248, 2724, 3180, ...
n | bases such that there is unique prime with period n

|

1 | 3, 4, 5, 6, 8, 10, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278, 282, 284, 294, 308, 312, 314, 318, 332, 338, 348, 350, 354, 360, 368, 374, 380, 384, 390, 398, 402, 410, 420, 422, 432, 434, 440, 444, 450, 458, 462, 464, 468, 480, 488, 492, 500, 504, 510, 522, 524, 542, 548, 558, 564, 570, 572, 578, 588, 594, 600, 602, 608, 614, 618, 620, 632, 642, 644, 648, 654, 660, 662, 674, 678, 684, 692, 702, 710, 720, 728, 734, 740, 744, 752, 758, 762, 770, 774, 788, 798, 810, 812, 822, 824, 828, 830, 840, 854, 858, 860, 864, 878, 882, 884, 888, 908, 912, 920, 930, 938, 942, 948, 954, 968, 972, 978, 984, 992, 998, 1010, 1014, 1020, 1022, 1032, 1034, 1040, 1050, 1052, 1062, 1064, 1070, 1088, 1092, 1094, 1098, 1104, 1110, 1118, 1124, 1130, 1152, 1154, 1164, 1172, 1182, 1188, 1194, 1202, 1214, 1218, 1224, 1230, 1232, 1238, 1250, 1260, 1278, 1280, 1284, 1290, 1292, 1298, 1302, 1304, 1308, 1320, 1322, 1328, 1362, 1368, 1374, 1382, 1400, 1410, 1424, 1428, 1430, 1434, 1440, 1448, 1452, 1454, 1460, 1472, 1482, 1484, 1488, 1490, 1494, 1500, 1512, 1524, 1532, 1544, 1550, 1554, 1560, 1568, 1572, 1580, 1584, 1598, 1602, 1608, 1610, 1614, 1620, 1622, 1628, 1638, 1658, 1664, 1668, 1670, 1694, 1698, 1700, 1710, 1722, 1724, 1734, 1742, 1748, 1754, 1760, 1778, 1784, 1788, 1790, 1802, 1812, 1824, 1832, 1848, 1862, 1868, 1872, 1874, 1878, 1880, 1890, 1902, 1908, 1914, 1932, 1934, 1950, 1952, 1974, 1980, 1988, 1994, 1998, 2000, 2004, 2012, 2018, 2028, 2030, 2040, 2054, 2064, 2070, 2082, 2084, 2088, 2090, 2100, 2112, 2114, 2130, 2132, 2138, 2142, 2144, 2154, 2162, 2180, 2204, 2208, 2214, 2222, 2238, 2240, 2244, 2252, 2268, 2270, 2274, 2282, 2288, 2294, 2298, 2310, 2312, 2334, 2340, 2342, 2348, 2352, 2358, 2372, 2378, 2382, 2384, 2390, 2394, 2400, 2412, 2418, 2424, 2438, 2442, 2448, 2460, 2468, 2474, 2478, ...
2 | 2, 4, 5, 6, 9, 10, 11, 12, 13, 16, 18, 19, 21, 22, 23, 25, 27, 28, 30, 33, 36, 37, 39, 40, 42, 43, 45, 46, 47, 51, 52, 55, 57, 58, 60, 61, 66, 67, 70, 72, 73, 75, 78, 79, 81, 82, 85, 87, 88, 91, 93, 95, 96, 100, 102, 103, 105, 106, 108, 111, 112, 115, 117, 121, 123, 126, 130, 133, 135, 136, 138, 141, 145, 147, 148, 150, 151, 156, 157, 159, 162, 163, 165, 166, 171, 172, 175, 177, 178, 180, 183, 187, 190, 191, 192, 193, 196, 198, 201, 205, 207, 210, 211, 213, 217, 222, 223, 225, 226, 228, 231, 232, 235, 238, 240, 243, 247, 250, 253, 256, 261, 262, 267, 268, 270, 271, 273, 276, 277, 280, 282, 283, 291, 292, 295, 297, 301, 303, 306, 310, 312, 313, 315, 316, 319, 325, 327, 330, 331, 333, 336, 343, 345, 346, 348, 351, 352, 355, 357, 358, 361, 366, 367, 372, 375, 378, 381, 382, 383, 385, 387, 388, 393, 396, 397, 400, 403, 408, 411, 415, 418, 420, 421, 423, 427, 430, 432, 435, 438, 442, 445, 447, 448, 451, 453, 456, 457, 460, 462, 463, 465, 466, 471, 477, 478, 481, 486, 487, 490, 495, 498, 501, 502, 507, 508, 513, 520, 522, 523, 525, 535, 537, 540, 541, 543, 546, 547, 553, 555, 556, 561, 562, 565, 567, 568, 570, 576, 583, 585, 586, 591, 592, 595, 598, 600, 603, 606, 607, 612, 613, 616, 618, 621, 625, 627, 630, 631, 633, 639, 640, 642, 646, 651, 652, 655, 658, 660, 661, 663, 667, 672, 673, 676, 682, 687, 690, 691, 693, 697, 700, 703, 705, 708, 711, 715, 717, 718, 723, 726, 732, 733, 735, 738, 742, 745, 750, 751, 756, 757, 760, 763, 765, 767, 768, 771, 772, 775, 777, 786, 787, 793, 795, 796, 801, 807, 808, 810, 817, 820, 822, 823, 826, 828, 831, 837, 838, 841, 843, 847, 852, 855, 856, 858, 861, 862, 865, 871, 876, 877, 880, 882, 885, 886, 891, 895, 897, 903, 906, 907, 910, 913, 915, 918, 921, 925, 927, 928, 931, 933, 936, 940, 943, 946, 952, 955, 957, 963, 966, 970, 973, 975, 976, 981, 982, 990, 991, 996, 997, 1003, 1005, 1008, 1012, 1015, 1017, 1018, 1020, 1027, 1030, 1032, 1038, 1041, 1045, 1047, 1048, 1050, 1051, 1060, 1062, 1068, 1071, 1075, 1081, 1083, 1086, 1087, 1090, 1092, 1093, 1095, 1096, 1102, 1107, 1108, 1111, 1113, 1116, 1122, 1123, 1125, 1128, 1131, 1135, 1137, 1141, 1150, 1152, 1153, 1162, 1167, 1170, 1171, 1173, 1180, 1183, 1185, 1186, 1191, 1192, 1197, 1200, 1201, 1207, 1212, 1213, 1215, 1216, 1222, 1225, 1227, 1228, 1230, 1233, 1236, 1237, 1243, 1248, 1251, 1255, 1258, 1261, 1263, 1267, 1276, 1278, 1279, 1281, 1282, 1285, 1288, 1290, 1293, 1296, 1300, 1302, 1303, 1305, 1306, 1311, 1317, 1318, 1320, 1321, 1323, 1326, 1327, 1335, 1345, 1347, 1353, 1360, 1365, 1366, 1372, 1375, 1380, 1381, 1383, 1387, 1395, 1398, 1401, 1407, 1408, 1411, 1417, 1422, 1423, 1426, 1428, 1431, 1432, 1435, 1437, 1438, 1446, 1447, 1450, 1452, 1453, 1458, 1465, 1467, 1470, 1471, 1477, 1480, 1482, 1485, 1486, 1488, 1491, 1492, 1498, 1501, 1503, 1510, 1513, 1515, 1521, 1522, 1527, 1530, 1531, 1535, 1537, 1542, 1543, 1545, 1548, 1551, 1552, 1555, 1558, 1566, 1570, 1573, 1575, 1578, 1582, 1587, 1591, 1593, 1596, 1600, 1603, 1606, 1608, 1612, 1615, 1617, 1618, 1620, 1621, 1626, 1635, 1636, 1641, 1645, 1647, 1653, 1656, 1657, 1662, 1663, 1666, 1668, 1675, 1677, 1683, 1687, 1692, 1695, 1696, 1698, 1705, 1708, 1711, 1713, 1717, 1720, 1722, 1723, 1725, 1731, 1732, 1740, 1743, 1746, 1752, 1753, 1755, 1758, 1761, 1765, 1771, 1773, 1776, 1782, 1783, 1786, 1788, 1791, 1795, 1800, 1807, 1810, 1813, 1815, 1821, 1822, 1827, 1830, 1831, 1837, 1843, 1846, 1851, 1855, 1857, 1860, 1863, 1866, 1867, 1870, 1872, 1873, 1876, 1878, 1881, 1887, 1888, 1893, 1900, 1905, 1906, 1911, 1912, 1915, 1927, 1930, 1932, 1933, 1941, 1947, 1948, 1950, 1951, 1953, 1963, 1965, 1972, 1978, 1981, 1983, 1986, 1992, 1993, 1995, 1996, 1998, 2002, 2007, 2010, 2011, 2016, 2017, 2025, 2026, 2028, 2031, 2035, 2037, 2038, 2041, 2052, 2055, 2061, 2062, 2065, 2068, 2077, 2080, 2082, 2083, 2086, 2088, 2091, 2095, 2097, 2098, 2101, 2103, 2110, 2112, 2121, 2125, 2128, 2130, 2136, 2137, 2140, 2142, 2143, 2151, 2152, 2160, 2163, 2167, 2173, 2175, 2178, 2181, 2185, 2187, 2191, 2193, 2202, 2205, 2206, 2212, 2215, 2217, 2220, 2223, 2227, 2233, 2236, 2238, 2242, 2245, 2247, 2250, 2251, 2257, 2263, 2266, 2268, 2271, 2272, 2275, 2280, 2283, 2286, 2292, 2296, 2301, 2305, 2307, 2308, 2310, 2325, 2332, 2335, 2338, 2340, 2341, 2343, 2346, 2347, 2350, 2356, 2361, 2367, 2370, 2371, 2373, 2376, 2380, 2382, 2383, 2385, 2388, 2392, 2395, 2398, 2401, 2403, 2410, 2415, 2416, 2422, 2425, 2427, 2431, 2433, 2436, 2440, 2445, 2446, 2451, 2455, 2457, 2458, 2461, 2466, 2467, 2472, 2473, 2475, 2476, 2487, 2497, ...
3 | 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 14, 15, 17, 19, 20, 21, 24, 27, 28, 31, 33, 34, 38, 40, 41, 43, 50, 52, 54, 57, 59, 62, 66, 69, 70, 71, 73, 75, 76, 77, 78, 80, 82, 85, 89, 90, 91, 97, 99, 101, 103, 105, 110, 111, 112, 115, 117, 119, 124, 127, 131, 136, 138, 141, 143, 145, 147, 148, 150, 153, 155, 157, 161, 162, 164, 166, 167, 168, 173, 175, 176, 187, 188, 189, 190, 192, 194, 199, 202, 203, 206, 209, 215, 218, 223, 231, 236, 241, 244, 245, 246, 259, 265, 266, 271, 272, 274, 278, 279, 280, 286, 287, 288, 290, 293, 309, 314, 316, 325, 329, 332, 336, 342, 344, 348, 351, 357, 358, 369, 370, 376, 378, 381, 383, 385, 388, 392, 395, 398, 402, 404, 405, 409, 414, 416, 421, 426, 427, 434, 435, 442, 447, 453, 455, 456, 460, 469, 472, 475, 476, 489, 493, 495, 496, 500, 511, 512, 514, 517, 518, 525, 526, 530, 531, 533, 537, 538, 540, 551, 553, 554, 556, 559, 560, 566, 567, 572, 574, 579, 580, 582, 584, 586, 589, 603, 605, 609, 612, 616, 621, 622, 624, 626, 628, 635, 637, 642, 644, 649, 658, 661, 668, 671, 673, 677, 682, 686, 696, 700, 701, 712, 715, 720, 726, 727, 728, 735, 736, 743, 747, 754, 755, 761, 762, 768, 773, 775, 778, 782, 785, 792, 798, 799, 801, 812, 818, 819, 825, 827, 832, 836, 838, 839, 846, 850, 855, 857, 859, 860, 864, 868, 875, 878, 883, 889, 890, 894, 895, 897, 899, 910, 911, 915, 916, 918, 920, 922, 925, 927, 937, 943, 946, 950, 959, 960, 964, 969, 974, 981, 987, 990, 992, 993, 994, 1000, 1001, 1002, 1006, 1007, 1011, 1016, 1020, 1021, 1022, 1027, 1029, 1039, 1041, 1051, 1057, 1058, 1060, 1063, 1065, 1067, 1070, 1072, 1074, 1077, 1078, 1091, 1092, 1093, 1097, 1098, 1105, 1107, 1116, 1118, 1125, 1126, 1130, 1133, 1142, 1144, 1146, 1148, 1149, 1163, 1165, 1174, 1181, 1182, 1191, 1193, 1195, 1196, 1198, 1200, 1202, 1209, 1216, 1217, 1221, 1230, 1252, 1254, 1256, 1258, 1260, 1263, 1272, 1274, 1275, 1281, 1288, 1295, 1308, 1312, 1314, 1317, 1323, 1324, 1330, 1331, 1333, 1336, 1340, 1343, 1345, 1349, 1351, 1352, 1354, 1359, 1363, 1364, 1365, 1372, 1373, 1380, 1386, 1392, 1396, 1401, 1405, 1414, 1415, 1421, 1422, 1427, 1434, 1438, 1440, 1441, 1442, 1443, 1447, 1448, 1449, 1454, 1456, 1473, 1475, 1483, 1484, 1487, 1492, 1494, 1505, 1510, 1515, 1519, 1520, 1522, 1526, 1529, 1534, 1536, 1539, 1540, 1547, 1548, 1559, 1560, 1561, 1562, 1566, 1568, 1573, 1581, 1583, 1585, 1590, 1592, 1594, 1596, 1609, 1611, 1616, 1620, 1624, 1630, 1632, 1639, 1644, 1646, 1650, 1651, 1657, 1658, 1659, 1665, 1666, 1669, 1672, 1676, 1678, 1685, 1690, 1692, 1695, 1700, 1701, 1707, 1708, 1709, 1711, 1716, 1718, 1720, 1723, 1729, 1746, 1748, 1750, 1753, 1757, 1762, 1767, 1770, 1788, 1795, 1800, 1804, 1806, 1809, 1811, 1814, 1818, 1825, 1826, 1828, 1832, 1837, 1841, 1847, 1848, 1853, 1861, 1863, 1879, 1882, 1886, 1895, 1902, 1909, 1923, 1931, 1932, 1939, 1952, 1956, 1960, 1963, 1970, 1973, 1974, 1977, 1982, 1986, 1993, 1994, 1996, 2000, 2010, 2015, 2016, 2022, 2029, 2033, 2043, 2049, 2054, 2056, 2058, 2064, 2065, 2068, 2073, 2075, 2079, 2080, 2085, 2086, 2091, 2105, 2107, 2110, 2112, 2117, 2129, 2136, 2138, 2140, 2142, 2145, 2157, 2159, 2162, 2163, 2168, 2170, 2171, 2176, 2183, 2191, 2194, 2197, 2201, 2203, 2204, 2205, 2227, 2233, 2236, 2238, 2241, 2247, 2255, 2259, 2260, 2262, 2264, 2273, 2275, 2276, 2280, 2290, 2296, 2301, 2303, 2309, 2313, 2318, 2324, 2339, 2345, 2353, 2359, 2360, 2364, 2372, 2373, 2380, 2385, 2387, 2390, 2397, 2402, 2409, 2411, 2422, 2429, 2430, 2435, 2437, 2442, 2446, 2450, 2451, 2455, 2456, 2463, 2464, 2465, 2470, 2471, 2476, 2478, 2483, 2484, 2491, 2493, 2495, 2497, 2498, ...
4 | 2, 3, 4, 5, 6, 9, 10, 11, 14, 15, 16, 19, 20, 24, 25, 26, 29, 35, 36, 39, 40, 45, 49, 51, 54, 56, 59, 61, 65, 66, 69, 71, 74, 79, 84, 85, 90, 94, 95, 101, 110, 116, 120, 121, 124, 126, 130, 131, 134, 139, 141, 145, 146, 150, 156, 159, 160, 165, 169, 170, 171, 175, 176, 180, 181, 184, 195, 199, 201, 204, 205, 206, 209, 210, 219, 221, 224, 230, 231, 236, 240, 245, 250, 256, 260, 261, 264, 270, 271, 275, 279, 280, 284, 289, 299, 300, 306, 309, 314, 315, 321, 325, 326, 329, 335, 340, 345, 349, 350, 371, 375, 379, 384, 386, 391, 396, 399, 400, 405, 406, 409, 415, 420, 425, 430, 435, 436, 440, 441, 444, 445, 449, 451, 459, 461, 464, 466, 470, 471, 474, 490, 496, 519, 521, 529, 535, 536, 544, 545, 556, 559, 569, 570, 571, 575, 576, 579, 581, 584, 594, 595, 609, 631, 634, 636, 639, 641, 644, 646, 649, 654, 661, 669, 674, 680, 685, 686, 689, 690, 695, 696, 699, 700, 704, 711, 714, 715, 716, 739, 740, 745, 750, 751, 760, 764, 779, 780, 781, 784, 791, 799, 815, 816, 819, 821, 826, 841, 855, 860, 861, 864, 869, 875, 881, 885, 890, 901, 906, 909, 910, 920, 921, 925, 929, 930, 935, 936, 946, 949, 950, 951, 955, 959, 960, 966, 979, 981, 985, 986, 989, 991, 1001, 1004, 1010, 1011, 1025, 1029, 1031, 1036, 1039, 1051, 1054, 1055, 1060, 1066, 1069, 1070, 1080, 1081, 1091, 1094, 1095, 1096, 1099, 1106, 1111, 1124, 1125, 1129, 1140, 1144, 1146, 1150, 1151, 1155, 1156, 1161, 1171, 1174, 1176, 1179, 1181, 1184, 1185, 1199, 1205, 1210, 1219, 1225, 1241, 1244, 1246, 1251, 1255, 1265, 1274, 1276, 1281, 1285, 1290, 1294, 1299, 1306, 1311, 1314, 1315, 1316, 1320, 1324, 1329, 1340, 1345, 1349, 1350, 1354, 1359, 1361, 1366, 1374, 1376, 1389, 1391, 1394, 1405, 1406, 1410, 1411, 1416, 1419, 1420, 1421, 1430, 1434, 1439, 1440, 1456, 1459, 1460, 1465, 1469, 1489, 1494, 1495, 1499, 1501, 1504, 1515, 1519, 1521, 1524, 1531, 1535, 1541, 1545, 1546, 1550, 1556, 1564, 1566, 1569, 1570, 1576, 1580, 1586, 1589, 1599, 1605, 1614, 1615, 1616, 1629, 1635, 1639, 1640, 1644, 1654, 1660, 1661, 1664, 1674, 1676, 1684, 1686, 1689, 1691, 1700, 1701, 1705, 1709, 1715, 1716, 1736, 1741, 1749, 1756, 1765, 1766, 1771, 1774, 1784, 1790, 1794, 1805, 1811, 1816, 1824, 1831, 1845, 1850, 1855, 1859, 1860, 1869, 1870, 1875, 1876, 1884, 1894, 1899, 1901, 1910, 1915, 1920, 1935, 1940, 1949, 1964, 1966, 1970, 1974, 1980, 1990, 2001, 2006, 2009, 2021, 2026, 2034, 2035, 2050, 2051, 2054, 2055, 2056, 2064, 2065, 2074, 2079, 2080, 2084, 2086, 2091, 2094, 2096, 2106, 2116, 2120, 2125, 2126, 2135, 2136, 2139, 2145, 2151, 2154, 2159, 2161, 2165, 2171, 2174, 2175, 2199, 2201, 2210, 2224, 2239, 2255, 2260, 2261, 2266, 2269, 2271, 2281, 2285, 2286, 2289, 2294, 2304, 2310, 2314, 2315, 2320, 2321, 2326, 2330, 2331, 2336, 2341, 2349, 2351, 2354, 2360, 2375, 2379, 2380, 2381, 2389, 2391, 2404, 2405, 2406, 2420, 2425, 2430, 2454, 2456, 2459, 2460, 2464, 2470, 2481, 2496, ...
5 | 2, 3, 6, 7, 11, 12, 13, 17, 21, 22, 23, 24, 28, 29, 30, 40, 41, 43, 44, 46, 50, 56, 62, 63, 68, 73, 74, 77, 79, 83, 85, 94, 99, 110, 111, 117, 118, 120, 122, 127, 129, 134, 143, 145, 151, 154, 162, 164, 165, 166, 172, 175, 177, 186, 193, 198, 204, 206, 208, 222, 227, 239, 249, 254, 255, 260, 261, 263, 265, 274, 275, 276, 277, 285, 288, 292, 304, 308, 327, 336, 337, 340, 352, 359, 369, 373, 393, 397, 408, 414, 417, 418, 437, 439, 448, 457, 459, 474, 479, 486, 490, 492, 495, 496, 503, 505, 514, 519, 528, 530, 536, 538, 539, 540, 550, 556, 557, 561, 563, 567, 568, 572, 579, 594, 601, 604, 616, 617, 626, 637, 645, 650, 662, 679, 694, 699, 706, 711, 714, 728, 745, 750, 761, 765, 770, 772, 793, 804, 805, 824, 826, 837, 854, 860, 864, 868, 880, 886, 890, 891, 919, 921, 931, 936, 941, 942, 954, 967, 968, 974, 976, 979, 986, 991, 1001, 1007, 1009, 1044, 1045, 1047, 1053, 1066, 1078, 1079, 1085, 1102, 1108, 1111, 1129, 1133, 1135, 1152, 1154, 1157, 1173, 1178, 1183, 1187, 1201, 1209, 1228, 1239, 1249, 1251, 1253, 1266, 1276, 1282, 1284, 1297, 1327, 1333, 1343, 1348, 1352, 1370, 1377, 1382, 1396, 1398, 1416, 1419, 1426, 1429, 1452, 1462, 1463, 1475, 1480, 1481, 1493, 1503, 1506, 1520, 1525, 1536, 1553, 1557, 1562, 1563, 1579, 1596, 1641, 1646, 1657, 1662, 1663, 1668, 1672, 1679, 1683, 1684, 1694, 1695, 1696, 1704, 1746, 1748, 1749, 1755, 1760, 1762, 1767, 1770, 1788, 1826, 1836, 1838, 1843, 1859, 1881, 1883, 1891, 1914, 1921, 1927, 1944, 1947, 1960, 1975, 1977, 1979, 1981, 1987, 1990, 1991, 2021, 2032, 2034, 2037, 2041, 2052, 2059, 2069, 2076, 2080, 2086, 2090, 2097, 2098, 2111, 2118, 2123, 2131, 2134, 2164, 2167, 2175, 2177, 2218, 2228, 2233, 2243, 2245, 2252, 2261, 2272, 2274, 2289, 2295, 2299, 2300, 2305, 2318, 2321, 2328, 2332, 2340, 2356, 2361, 2362, 2366, 2373, 2375, 2382, 2383, 2387, 2430, 2431, 2432, 2438, 2448, 2450, 2464, 2483, 2487, 2499, ...
6 | 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 16, 18, 20, 21, 22, 25, 28, 29, 32, 34, 35, 39, 41, 42, 44, 51, 53, 55, 58, 60, 63, 67, 70, 71, 72, 74, 76, 77, 78, 79, 81, 83, 86, 90, 91, 92, 98, 100, 102, 104, 106, 111, 112, 113, 116, 118, 120, 125, 128, 132, 137, 139, 142, 144, 146, 148, 149, 151, 154, 156, 158, 162, 163, 165, 167, 168, 169, 174, 176, 177, 188, 189, 190, 191, 193, 195, 200, 203, 204, 207, 210, 216, 219, 224, 232, 237, 242, 245, 246, 247, 260, 266, 267, 272, 273, 275, 279, 280, 281, 287, 288, 289, 291, 294, 310, 315, 317, 326, 330, 333, 337, 343, 345, 349, 352, 358, 359, 370, 371, 377, 379, 382, 384, 386, 389, 393, 396, 399, 403, 405, 406, 410, 415, 417, 422, 427, 428, 435, 436, 443, 448, 454, 456, 457, 461, 470, 473, 476, 477, 490, 494, 496, 497, 501, 512, 513, 515, 518, 519, 526, 527, 531, 532, 534, 538, 539, 541, 552, 554, 555, 557, 560, 561, 567, 568, 573, 575, 580, 581, 583, 585, 587, 590, 604, 606, 610, 613, 617, 622, 623, 625, 627, 629, 636, 638, 643, 645, 650, 659, 662, 669, 672, 674, 678, 683, 687, 697, 701, 702, 713, 716, 721, 727, 728, 729, 736, 737, 744, 748, 755, 756, 762, 763, 769, 774, 776, 779, 783, 786, 793, 799, 800, 802, 813, 819, 820, 826, 828, 833, 837, 839, 840, 847, 851, 856, 858, 860, 861, 865, 869, 876, 879, 884, 890, 891, 895, 896, 898, 900, 911, 912, 916, 917, 919, 921, 923, 926, 928, 938, 944, 947, 951, 960, 961, 965, 970, 975, 982, 988, 991, 993, 994, 995, 1001, 1002, 1003, 1007, 1008, 1012, 1017, 1021, 1022, 1023, 1028, 1030, 1040, 1042, 1052, 1058, 1059, 1061, 1064, 1066, 1068, 1071, 1073, 1075, 1078, 1079, 1092, 1093, 1094, 1098, 1099, 1106, 1108, 1117, 1119, 1126, 1127, 1131, 1134, 1143, 1145, 1147, 1149, 1150, 1164, 1166, 1175, 1182, 1183, 1192, 1194, 1196, 1197, 1199, 1201, 1203, 1210, 1217, 1218, 1222, 1231, 1253, 1255, 1257, 1259, 1261, 1264, 1273, 1275, 1276, 1282, 1289, 1296, 1309, 1313, 1315, 1318, 1324, 1325, 1331, 1332, 1334, 1337, 1341, 1344, 1346, 1350, 1352, 1353, 1355, 1360, 1364, 1365, 1366, 1373, 1374, 1381, 1387, 1393, 1397, 1402, 1406, 1415, 1416, 1422, 1423, 1428, 1435, 1439, 1441, 1442, 1443, 1444, 1448, 1449, 1450, 1455, 1457, 1474, 1476, 1484, 1485, 1488, 1493, 1495, 1506, 1511, 1516, 1520, 1521, 1523, 1527, 1530, 1535, 1537, 1540, 1541, 1548, 1549, 1560, 1561, 1562, 1563, 1567, 1569, 1574, 1582, 1584, 1586, 1591, 1593, 1595, 1597, 1610, 1612, 1617, 1621, 1625, 1631, 1633, 1640, 1645, 1647, 1651, 1652, 1658, 1659, 1660, 1666, 1667, 1670, 1673, 1677, 1679, 1686, 1691, 1693, 1696, 1701, 1702, 1708, 1709, 1710, 1712, 1717, 1719, 1721, 1724, 1730, 1747, 1749, 1751, 1754, 1758, 1763, 1768, 1771, 1789, 1796, 1801, 1805, 1807, 1810, 1812, 1815, 1819, 1826, 1827, 1829, 1833, 1838, 1842, 1848, 1849, 1854, 1862, 1864, 1880, 1883, 1887, 1896, 1903, 1910, 1924, 1932, 1933, 1940, 1953, 1957, 1961, 1964, 1971, 1974, 1975, 1978, 1983, 1987, 1994, 1995, 1997, 2001, 2011, 2016, 2017, 2023, 2030, 2034, 2044, 2050, 2055, 2057, 2059, 2065, 2066, 2069, 2074, 2076, 2080, 2081, 2086, 2087, 2092, 2106, 2108, 2111, 2113, 2118, 2130, 2137, 2139, 2141, 2143, 2146, 2158, 2160, 2163, 2164, 2169, 2171, 2172, 2177, 2184, 2192, 2195, 2198, 2202, 2204, 2205, 2206, 2228, 2234, 2237, 2239, 2242, 2248, 2256, 2260, 2261, 2263, 2265, 2274, 2276, 2277, 2281, 2291, 2297, 2302, 2304, 2310, 2314, 2319, 2325, 2340, 2346, 2354, 2360, 2361, 2365, 2373, 2374, 2381, 2386, 2388, 2391, 2398, 2403, 2410, 2412, 2423, 2430, 2431, 2436, 2438, 2443, 2447, 2451, 2452, 2456, 2457, 2464, 2465, 2466, 2471, 2472, 2477, 2479, 2484, 2485, 2492, 2494, 2496, 2498, 2499, ...
7 | 2, 3, 5, 6, 13, 14, 15, 17, 22, 26, 29, 31, 38, 40, 46, 50, 56, 60, 61, 66, 68, 72, 73, 80, 85, 87, 89, 93, 95, 115, 120, 122, 126, 128, 134, 146, 149, 156, 158, 160, 163, 180, 186, 192, 203, 206, 208, 220, 221, 235, 237, 238, 246, 251, 264, 266, 267, 280, 282, 290, 294, 300, 303, 309, 320, 330, 341, 349, 350, 351, 353, 363, 381, 395, 399, 404, 405, 414, 417, 418, 421, 436, 438, 447, 450, 461, 464, 466, 470, 478, 505, 519, 523, 531, 533, 539, 548, 560, 583, 584, 589, 591, 599, 609, 611, 622, 638, 646, 647, 655, 657, 660, 681, 698, 700, 701, 710, 717, 734, 760, 765, 776, 798, 800, 802, 805, 820, 822, 842, 855, 856, 862, 863, 869, 870, 878, 899, 911, 912, 913, 926, 927, 931, 940, 941, 942, 947, 959, 984, 998, 1016, 1023, 1029, 1046, 1052, 1055, 1081, 1084, 1087, 1111, 1115, 1117, 1131, 1136, 1144, 1148, 1161, 1169, 1170, 1188, 1189, 1194, 1195, 1197, 1221, 1224, 1229, 1248, 1266, 1277, 1280, 1281, 1284, 1291, 1297, 1309, 1316, 1333, 1339, 1348, 1363, 1364, 1371, 1377, 1382, 1402, 1404, 1409, 1418, 1419, 1424, 1425, 1436, 1438, 1439, 1455, 1469, 1471, 1480, 1482, 1489, 1494, 1512, 1513, 1520, 1529, 1537, 1547, 1550, 1554, 1558, 1563, 1565, 1566, 1567, 1572, 1579, 1585, 1596, 1604, 1605, 1606, 1608, 1609, 1622, 1627, 1630, 1651, 1668, 1672, 1679, 1687, 1700, 1708, 1710, 1714, 1716, 1738, 1750, 1753, 1757, 1758, 1762, 1766, 1772, 1783, 1790, 1796, 1797, 1800, 1802, 1808, 1816, 1831, 1835, 1839, 1840, 1854, 1855, 1885, 1888, 1899, 1904, 1911, 1915, 1918, 1920, 1929, 1940, 1941, 1958, 1978, 1985, 1990, 1998, 2015, 2039, 2057, 2071, 2072, 2087, 2094, 2097, 2100, 2115, 2126, 2127, 2131, 2139, 2151, 2152, 2156, 2157, 2158, 2159, 2164, 2181, 2186, 2188, 2190, 2193, 2201, 2205, 2219, 2222, 2226, 2230, 2232, 2244, 2245, 2265, 2274, 2293, 2299, 2303, 2306, 2308, 2321, 2359, 2378, 2404, 2408, 2409, 2416, 2436, 2437, 2440, 2450, 2458, 2463, 2466, 2480, 2491, 2493, 2496, ...
8 | 2, 3, 4, 5, 6, 7, 11, 13, 16, 17, 20, 21, 23, 24, 28, 29, 34, 35, 39, 46, 48, 54, 56, 57, 61, 65, 71, 73, 74, 80, 81, 82, 88, 90, 103, 105, 106, 113, 115, 118, 119, 129, 132, 140, 142, 153, 154, 160, 164, 165, 169, 171, 174, 180, 194, 198, 199, 203, 204, 205, 210, 220, 228, 238, 242, 248, 251, 254, 259, 266, 267, 272, 275, 276, 278, 288, 296, 309, 312, 313, 317, 320, 328, 333, 334, 337, 339, 340, 352, 353, 363, 364, 374, 403, 405, 414, 415, 419, 430, 431, 436, 442, 445, 449, 453, 455, 463, 466, 471, 472, 476, 477, 479, 487, 488, 492, 494, 498, 503, 504, 505, 513, 516, 517, 523, 526, 537, 539, 540, 543, 550, 551, 554, 556, 561, 566, 567, 568, 573, 579, 582, 584, 600, 605, 607, 613, 616, 623, 624, 628, 639, 643, 649, 656, 657, 677, 681, 690, 701, 702, 703, 710, 713, 719, 721, 725, 730, 732, 738, 742, 745, 748, 758, 760, 761, 768, 769, 772, 778, 786, 788, 795, 798, 800, 805, 810, 811, 819, 821, 829, 833, 843, 845, 855, 856, 857, 874, 879, 883, 891, 894, 895, 912, 913, 914, 917, 919, 928, 930, 931, 936, 952, 962, 963, 965, 966, 986, 992, 996, 997, 1007, 1013, 1015, 1019, 1038, 1041, 1042, 1047, 1049, 1053, 1057, 1071, 1072, 1076, 1087, 1088, 1089, 1101, 1119, 1123, 1125, 1126, 1132, 1135, 1136, 1142, 1144, 1150, 1152, 1163, 1170, 1173, 1180, 1183, 1200, 1202, 1218, 1236, 1238, 1246, 1252, 1253, 1259, 1269, 1270, 1275, 1280, 1302, 1305, 1322, 1327, 1330, 1333, 1344, 1353, 1355, 1371, 1381, 1382, 1383, 1388, 1389, 1401, 1404, 1406, 1428, 1434, 1441, 1442, 1446, 1457, 1458, 1463, 1472, 1483, 1485, 1486, 1493, 1496, 1527, 1529, 1533, 1536, 1540, 1542, 1547, 1552, 1554, 1557, 1558, 1567, 1568, 1569, 1571, 1586, 1594, 1598, 1610, 1612, 1618, 1622, 1635, 1638, 1644, 1646, 1650, 1652, 1655, 1666, 1680, 1683, 1687, 1688, 1700, 1706, 1710, 1718, 1722, 1737, 1738, 1745, 1751, 1754, 1755, 1757, 1765, 1772, 1788, 1789, 1806, 1809, 1813, 1820, 1823, 1824, 1829, 1836, 1841, 1849, 1850, 1853, 1854, 1864, 1865, 1870, 1892, 1905, 1909, 1910, 1915, 1916, 1926, 1932, 1934, 1935, 1942, 1944, 1945, 1948, 1952, 1956, 1962, 1967, 1972, 1977, 1978, 1985, 1986, 1994, 2001, 2007, 2011, 2013, 2037, 2039, 2040, 2044, 2046, 2051, 2058, 2061, 2068, 2069, 2071, 2073, 2074, 2079, 2097, 2102, 2104, 2108, 2125, 2126, 2131, 2141, 2142, 2143, 2145, 2149, 2152, 2158, 2162, 2163, 2175, 2192, 2193, 2204, 2205, 2206, 2209, 2215, 2222, 2223, 2224, 2226, 2238, 2240, 2245, 2247, 2250, 2254, 2255, 2256, 2260, 2261, 2266, 2279, 2283, 2292, 2296, 2305, 2311, 2312, 2315, 2322, 2333, 2336, 2341, 2343, 2349, 2350, 2360, 2363, 2368, 2369, 2373, 2374, 2376, 2384, 2397, 2400, 2403, 2408, 2414, 2421, 2428, 2432, 2436, 2438, 2441, 2443, 2445, 2447, 2465, 2478, 2479, 2481, 2482, 2486, 2488, 2493, ...
9 | 2, 3, 8, 10, 11, 13, 20, 21, 26, 30, 50, 51, 56, 60, 67, 76, 78, 88, 91, 98, 102, 117, 129, 134, 146, 154, 159, 163, 171, 186, 189, 190, 191, 193, 198, 200, 209, 210, 212, 222, 240, 249, 250, 265, 267, 269, 278, 279, 299, 300, 307, 333, 344, 363, 383, 388, 390, 398, 399, 406, 425, 429, 436, 438, 439, 444, 450, 458, 496, 504, 509, 520, 523, 533, 540, 544, 570, 580, 585, 603, 634, 637, 646, 658, 666, 668, 691, 698, 704, 716, 721, 735, 748, 798, 806, 809, 818, 827, 844, 866, 875, 886, 887, 889, 894, 939, 952, 962, 984, 989, 998, 1001, 1002, 1003, 1021, 1025, 1029, 1040, 1041, 1044, 1064, 1072, 1075, 1102, 1121, 1139, 1152, 1153, 1186, 1188, 1198, 1207, 1224, 1227, 1234, 1235, 1247, 1264, 1268, 1276, 1280, 1285, 1286, 1287, 1300, 1305, 1314, 1331, 1349, 1364, 1399, 1400, 1408, 1409, 1420, 1427, 1432, 1445, 1447, 1473, 1484, 1512, 1519, 1527, 1557, 1560, 1569, 1577, 1590, 1591, 1609, 1614, 1618, 1626, 1630, 1641, 1649, 1654, 1664, 1682, 1685, 1687, 1692, 1704, 1706, 1717, 1739, 1741, 1761, 1769, 1774, 1787, 1805, 1806, 1813, 1826, 1843, 1850, 1865, 1870, 1873, 1907, 1910, 1915, 1937, 1949, 1950, 1959, 1976, 1984, 1990, 2016, 2021, 2029, 2040, 2059, 2062, 2065, 2073, 2082, 2085, 2112, 2124, 2136, 2139, 2141, 2160, 2167, 2174, 2200, 2203, 2223, 2243, 2257, 2276, 2287, 2288, 2298, 2302, 2307, 2339, 2345, 2350, 2357, 2363, 2390, 2395, 2397, 2405, 2406, 2424, 2428, 2440, 2463, 2466, 2469, 2477, 2480, 2497, ...
10 | 2, 3, 4, 5, 9, 10, 11, 12, 16, 20, 21, 22, 33, 34, 37, 38, 43, 47, 48, 55, 59, 69, 71, 75, 76, 80, 81, 99, 109, 111, 121, 124, 126, 131, 133, 135, 136, 141, 144, 155, 157, 158, 164, 165, 176, 177, 180, 199, 203, 223, 229, 242, 245, 251, 253, 254, 256, 257, 258, 264, 265, 268, 274, 276, 286, 290, 297, 307, 319, 322, 323, 334, 342, 361, 363, 366, 374, 375, 377, 385, 388, 396, 411, 427, 438, 439, 444, 452, 462, 483, 487, 489, 495, 499, 500, 505, 506, 507, 509, 510, 528, 531, 539, 549, 551, 555, 583, 584, 588, 603, 606, 609, 610, 628, 639, 641, 642, 652, 654, 658, 664, 676, 693, 702, 704, 707, 714, 716, 719, 738, 741, 749, 758, 760, 768, 774, 782, 791, 792, 812, 813, 815, 836, 839, 850, 861, 874, 894, 901, 912, 924, 935, 949, 960, 973, 978, 980, 982, 994, 1004, 1012, 1023, 1026, 1035, 1039, 1045, 1054, 1057, 1065, 1066, 1068, 1071, 1087, 1093, 1094, 1105, 1109, 1116, 1127, 1137, 1148, 1155, 1156, 1158, 1160, 1165, 1166, 1178, 1187, 1191, 1199, 1204, 1215, 1219, 1230, 1243, 1244, 1254, 1259, 1265, 1266, 1270, 1276, 1279, 1281, 1291, 1310, 1320, 1341, 1342, 1353, 1364, 1365, 1368, 1374, 1375, 1401, 1433, 1452, 1462, 1473, 1474, 1475, 1488, 1496, 1506, 1530, 1539, 1541, 1550, 1551, 1554, 1560, 1567, 1574, 1585, 1593, 1594, 1598, 1605, 1609, 1611, 1615, 1618, 1621, 1637, 1643, 1649, 1659, 1660, 1661, 1676, 1686, 1692, 1695, 1698, 1715, 1716, 1717, 1719, 1738, 1752, 1754, 1761, 1775, 1780, 1791, 1805, 1818, 1819, 1831, 1838, 1840, 1842, 1846, 1848, 1864, 1869, 1873, 1874, 1884, 1901, 1903, 1919, 1923, 1926, 1928, 1929, 1935, 1940, 1946, 1947, 1948, 1962, 1970, 1974, 1990, 1996, 2017, 2023, 2025, 2046, 2058, 2066, 2080, 2099, 2102, 2111, 2112, 2124, 2134, 2143, 2160, 2165, 2170, 2189, 2192, 2210, 2222, 2227, 2232, 2233, 2256, 2258, 2264, 2265, 2271, 2275, 2293, 2303, 2304, 2314, 2320, 2324, 2326, 2330, 2332, 2333, 2335, 2343, 2353, 2366, 2386, 2390, 2392, 2419, 2440, 2457, 2463, 2469, 2473, 2484, ...
11 | 5, 17, 20, 21, 23, 30, 53, 60, 86, 137, 172, 195, 212, 224, 229, 254, 258, 268, 272, 287, 298, 319, 339, 355, 365, 366, 389, 390, 398, 414, 467, 474, 480, 504, 534, 539, 543, 562, 567, 592, 619, 626, 654, 672, 709, 735, 756, 766, 770, 778, 787, 806, 812, 874, 943, 973, 1003, 1036, 1040, 1065, 1091, 1103, 1167, 1194, 1210, 1213, 1239, 1243, 1249, 1264, 1309, 1311, 1341, 1348, 1357, 1377, 1425, 1433, 1437, 1448, 1449, 1459, 1486, 1510, 1528, 1575, 1586, 1601, 1631, 1661, 1696, 1698, 1701, 1748, 1770, 1840, 1859, 1860, 1864, 1874, 1910, 1924, 1926, 1937, 1942, 1954, 1977, 1985, 1993, 2000, 2029, 2039, 2087, 2089, 2123, 2149, 2167, 2215, 2223, 2228, 2238, 2261, 2274, 2294, 2301, 2307, 2311, 2319, 2320, 2323, 2338, 2388, 2393, 2403, 2416, 2438, 2457, 2458, 2461, ...
12 | 2, 3, 4, 5, 9, 10, 12, 13, 17, 25, 27, 30, 31, 36, 38, 39, 43, 48, 52, 55, 56, 61, 62, 65, 83, 92, 94, 99, 100, 104, 105, 109, 114, 118, 126, 131, 166, 168, 169, 172, 183, 185, 190, 194, 196, 198, 209, 224, 225, 229, 231, 239, 244, 257, 260, 261, 263, 269, 270, 272, 278, 291, 296, 299, 300, 302, 308, 311, 313, 320, 324, 328, 329, 339, 343, 346, 348, 350, 363, 374, 376, 390, 407, 412, 413, 417, 424, 426, 429, 434, 437, 438, 443, 447, 455, 463, 468, 480, 482, 485, 494, 502, 503, 507, 511, 515, 525, 533, 534, 545, 549, 555, 558, 559, 568, 571, 582, 593, 595, 599, 610, 612, 614, 619, 625, 632, 638, 640, 641, 647, 653, 659, 676, 684, 686, 697, 699, 701, 702, 707, 710, 716, 718, 724, 725, 729, 738, 742, 749, 767, 768, 777, 780, 793, 794, 797, 798, 801, 805, 811, 820, 832, 836, 840, 842, 853, 854, 861, 862, 868, 871, 872, 876, 893, 905, 909, 915, 923, 924, 949, 953, 958, 959, 961, 963, 967, 972, 974, 978, 988, 1000, 1009, 1015, 1018, 1026, 1031, 1036, 1039, 1040, 1045, 1063, 1067, 1070, 1074, 1080, 1101, 1105, 1110, 1113, 1122, 1123, 1126, 1132, 1134, 1135, 1147, 1156, 1174, 1195, 1196, 1205, 1208, 1210, 1212, 1214, 1219, 1232, 1239, 1251, 1256, 1258, 1261, 1275, 1284, 1299, 1305, 1308, 1325, 1326, 1331, 1336, 1344, 1360, 1365, 1369, 1370, 1373, 1375, 1379, 1394, 1395, 1405, 1413, 1416, 1421, 1426, 1434, 1440, 1452, 1459, 1460, 1464, 1473, 1483, 1486, 1490, 1491, 1492, 1499, 1505, 1516, 1517, 1521, 1524, 1538, 1550, 1552, 1555, 1563, 1569, 1572, 1576, 1589, 1590, 1596, 1598, 1600, 1611, 1616, 1624, 1625, 1635, 1641, 1647, 1652, 1654, 1659, 1660, 1661, 1663, 1669, 1680, 1689, 1690, 1693, 1695, 1700, 1704, 1712, 1713, 1717, 1719, 1726, 1733, 1739, 1741, 1746, 1751, 1756, 1758, 1759, 1764, 1773, 1778, 1780, 1789, 1793, 1811, 1823, 1829, 1830, 1832, 1841, 1854, 1856, 1860, 1863, 1875, 1885, 1886, 1889, 1890, 1893, 1897, 1898, 1899, 1902, 1914, 1915, 1927, 1928, 1929, 1933, 1936, 1949, 1951, 1963, 1967, 1971, 1980, 1986, 1989, 2005, 2011, 2018, 2033, 2037, 2040, 2054, 2055, 2062, 2072, 2075, 2076, 2083, 2084, 2089, 2090, 2096, 2102, 2105, 2107, 2111, 2116, 2119, 2122, 2135, 2144, 2153, 2155, 2168, 2183, 2189, 2211, 2224, 2232, 2235, 2241, 2250, 2252, 2257, 2258, 2259, 2261, 2270, 2285, 2300, 2301, 2309, 2313, 2319, 2331, 2337, 2340, 2341, 2356, 2363, 2369, 2371, 2384, 2387, 2392, 2395, 2396, 2401, 2426, 2440, 2443, 2453, 2454, 2457, 2486, 2488, 2497, 2500, ...
13 | 2, 3, 5, 7, 34, 37, 43, 59, 72, 92, 94, 98, 110, 118, 133, 149, 151, 159, 190, 207, 219, 221, 251, 260, 264, 267, 274, 282, 286, 291, 319, 355, 363, 373, 382, 397, 398, 402, 406, 408, 412, 436, 442, 443, 486, 489, 507, 542, 544, 552, 553, 573, 582, 585, 592, 603, 610, 614, 634, 643, 645, 677, 689, 708, 720, 730, 744, 769, 772, 806, 851, 853, 862, 882, 912, 928, 930, 952, 968, 993, 1006, 1019, 1025, 1040, 1047, 1067, 1097, 1098, 1120, 1125, 1131, 1133, 1153, 1165, 1172, 1175, 1191, 1197, 1236, 1245, 1248, 1262, 1271, 1273, 1275, 1281, 1299, 1305, 1306, 1311, 1346, 1376, 1378, 1400, 1403, 1404, 1412, 1418, 1428, 1431, 1442, 1450, 1471, 1496, 1502, 1570, 1585, 1587, 1660, 1662, 1666, 1736, 1754, 1757, 1836, 1853, 1860, 1861, 1885, 1903, 1913, 1915, 1972, 1975, 1987, 1988, 1995, 2006, 2033, 2052, 2073, 2098, 2191, 2193, 2206, 2210, 2224, 2252, 2255, 2267, 2280, 2302, 2314, 2319, 2324, 2346, 2362, 2366, 2417, 2444, 2450, 2479, 2486, ...
14 | 2, 3, 10, 11, 14, 15, 16, 17, 18, 21, 24, 25, 29, 37, 43, 44, 46, 48, 49, 52, 54, 61, 66, 69, 72, 73, 78, 84, 86, 87, 99, 101, 104, 106, 114, 115, 128, 133, 135, 136, 143, 145, 146, 148, 164, 169, 170, 173, 200, 219, 224, 226, 228, 231, 234, 240, 244, 248, 255, 262, 275, 281, 282, 298, 301, 305, 320, 321, 322, 327, 330, 339, 347, 359, 363, 368, 369, 384, 385, 387, 401, 417, 433, 435, 442, 450, 455, 458, 482, 488, 494, 496, 508, 517, 519, 521, 553, 559, 566, 579, 582, 590, 601, 616, 621, 625, 628, 630, 636, 638, 654, 657, 669, 675, 691, 692, 694, 711, 712, 714, 719, 724, 725, 728, 732, 737, 750, 756, 757, 765, 768, 773, 774, 778, 780, 793, 797, 803, 810, 827, 828, 831, 842, 848, 860, 865, 869, 895, 901, 910, 916, 919, 922, 923, 926, 931, 936, 938, 951, 960, 967, 971, 972, 982, 1002, 1010, 1022, 1032, 1060, 1063, 1070, 1073, 1074, 1089, 1098, 1105, 1109, 1110, 1123, 1128, 1130, 1131, 1132, 1138, 1147, 1160, 1178, 1179, 1184, 1189, 1190, 1207, 1216, 1218, 1220, 1221, 1291, 1293, 1302, 1303, 1308, 1332, 1349, 1350, 1353, 1357, 1361, 1362, 1366, 1373, 1374, 1386, 1394, 1407, 1435, 1448, 1474, 1487, 1511, 1515, 1524, 1535, 1538, 1558, 1562, 1563, 1564, 1581, 1597, 1605, 1614, 1615, 1616, 1626, 1638, 1639, 1645, 1648, 1653, 1654, 1671, 1681, 1683, 1693, 1696, 1698, 1705, 1707, 1710, 1713, 1719, 1723, 1726, 1731, 1739, 1748, 1758, 1768, 1781, 1787, 1789, 1793, 1799, 1800, 1810, 1817, 1823, 1830, 1835, 1839, 1848, 1853, 1859, 1879, 1909, 1929, 1932, 1935, 1961, 1964, 1969, 1979, 1996, 1997, 2002, 2009, 2013, 2015, 2026, 2037, 2056, 2058, 2071, 2074, 2075, 2077, 2080, 2088, 2099, 2104, 2106, 2111, 2133, 2140, 2145, 2147, 2162, 2174, 2175, 2190, 2196, 2204, 2207, 2216, 2219, 2221, 2233, 2234, 2236, 2248, 2251, 2257, 2269, 2273, 2277, 2285, 2286, 2294, 2305, 2309, 2315, 2316, 2319, 2322, 2334, 2339, 2356, 2372, 2378, 2379, 2393, 2403, 2406, 2414, 2418, 2419, 2424, 2425, 2427, 2432, 2434, 2446, 2460, 2467, 2479, 2480, 2482, 2489, 2494, ...
15 | 2, 3, 11, 17, 23, 43, 46, 52, 53, 61, 62, 78, 84, 88, 89, 92, 99, 108, 123, 124, 141, 146, 154, 156, 158, 163, 170, 171, 182, 187, 202, 217, 219, 221, 229, 233, 238, 248, 249, 253, 264, 274, 275, 278, 283, 285, 287, 291, 296, 302, 309, 314, 315, 322, 325, 342, 346, 353, 356, 366, 368, 372, 377, 380, 384, 394, 404, 406, 411, 420, 425, 426, 446, 461, 463, 466, 470, 480, 482, 486, 487, 488, 494, 495, 498, 499, 508, 518, 521, 533, 540, 552, 554, 573, 580, 592, 593, 594, 595, 612, 624, 633, 636, 644, 647, 672, 681, 703, 707, 711, 712, 718, 719, 721, 724, 737, 740, 756, 760, 766, 791, 792, 801, 811, 821, 832, 852, 864, 872, 905, 914, 915, 938, 942, 946, 983, 984, 985, 990, 996, 1016, 1025, 1038, 1046, 1056, 1070, 1078, 1085, 1116, 1127, 1129, 1143, 1146, 1178, 1179, 1209, 1213, 1217, 1257, 1261, 1265, 1266, 1271, 1283, 1288, 1307, 1331, 1339, 1344, 1360, 1385, 1422, 1434, 1441, 1448, 1451, 1484, 1491, 1499, 1501, 1505, 1509, 1519, 1522, 1549, 1555, 1574, 1575, 1576, 1585, 1587, 1592, 1605, 1617, 1625, 1639, 1675, 1695, 1698, 1700, 1701, 1705, 1731, 1751, 1753, 1761, 1763, 1772, 1797, 1801, 1834, 1860, 1881, 1893, 1904, 1908, 1923, 1935, 1937, 1938, 1943, 1944, 1956, 1959, 1983, 1985, 2007, 2011, 2014, 2019, 2027, 2032, 2051, 2052, 2061, 2063, 2075, 2078, 2083, 2085, 2106, 2113, 2121, 2123, 2139, 2163, 2168, 2172, 2200, 2206, 2214, 2217, 2226, 2230, 2234, 2248, 2259, 2276, 2278, 2309, 2318, 2326, 2328, 2347, 2357, 2373, 2383, 2388, 2398, 2413, 2440, 2460, 2461, 2473, 2476, 2486, ...
16 | 2, 4, 9, 13, 33, 43, 47, 51, 53, 69, 81, 118, 132, 140, 145, 152, 185, 205, 208, 237, 239, 240, 242, 288, 290, 305, 306, 323, 341, 365, 373, 378, 392, 395, 409, 426, 433, 434, 442, 451, 455, 491, 501, 508, 510, 519, 531, 540, 542, 553, 557, 562, 565, 577, 596, 610, 664, 680, 682, 705, 723, 732, 747, 782, 795, 800, 808, 835, 841, 859, 866, 876, 884, 892, 916, 918, 934, 951, 956, 973, 990, 1021, 1022, 1028, 1037, 1053, 1054, 1103, 1106, 1113, 1120, 1121, 1135, 1143, 1174, 1199, 1205, 1209, 1211, 1224, 1232, 1239, 1256, 1284, 1307, 1310, 1343, 1351, 1375, 1377, 1379, 1390, 1403, 1446, 1458, 1475, 1477, 1479, 1495, 1505, 1513, 1529, 1531, 1580, 1585, 1602, 1614, 1616, 1623, 1628, 1640, 1681, 1682, 1691, 1698, 1699, 1732, 1749, 1752, 1753, 1755, 1777, 1785, 1800, 1815, 1836, 1837, 1845, 1868, 1871, 1878, 1885, 1930, 1936, 1942, 1947, 1991, 2053, 2061, 2078, 2083, 2090, 2100, 2104, 2117, 2123, 2140, 2159, 2175, 2191, 2193, 2194, 2227, 2257, 2261, 2274, 2279, 2282, 2286, 2291, 2297, 2328, 2330, 2337, 2359, 2362, 2378, 2405, 2429, 2440, 2467, 2480, 2490, ...
17 | 2, 11, 18, 20, 21, 28, 31, 55, 57, 62, 84, 87, 97, 107, 109, 129, 147, 149, 157, 160, 170, 181, 188, 189, 191, 205, 207, 241, 247, 251, 274, 295, 297, 315, 327, 335, 349, 351, 355, 358, 364, 365, 368, 379, 383, 410, 419, 423, 431, 436, 438, 466, 472, 506, 513, 527, 557, 571, 597, 599, 614, 637, 653, 656, 688, 708, 709, 720, 740, 762, 835, 836, 874, 974, 976, 980, 982, 986, 987, 1029, 1045, 1054, 1136, 1151, 1171, 1181, 1185, 1208, 1222, 1230, 1236, 1240, 1242, 1280, 1319, 1332, 1367, 1371, 1375, 1392, 1402, 1435, 1447, 1481, 1495, 1496, 1519, 1525, 1530, 1536, 1547, 1556, 1592, 1603, 1607, 1615, 1683, 1692, 1706, 1726, 1775, 1776, 1782, 1793, 1828, 1833, 1846, 1945, 1967, 1988, 1998, 2001, 2002, 2046, 2049, 2064, 2100, 2103, 2108, 2165, 2183, 2192, 2206, 2223, 2249, 2259, 2268, 2295, 2301, 2303, 2312, 2321, 2343, 2351, 2371, 2395, 2406, 2443, 2446, 2466, 2468, 2471, 2492, 2494, ...
18 | 2, 5, 6, 7, 8, 9, 11, 26, 35, 38, 42, 49, 54, 56, 66, 73, 80, 81, 88, 96, 100, 106, 107, 118, 120, 121, 122, 126, 131, 134, 175, 190, 196, 197, 207, 208, 227, 234, 235, 240, 245, 266, 271, 275, 296, 301, 302, 305, 308, 312, 322, 335, 342, 343, 347, 350, 353, 362, 367, 378, 379, 381, 384, 397, 399, 415, 418, 423, 442, 444, 455, 464, 467, 473, 480, 494, 505, 513, 520, 530, 556, 557, 574, 575, 581, 609, 628, 664, 665, 666, 689, 695, 700, 702, 708, 711, 712, 720, 723, 729, 739, 745, 795, 810, 823, 847, 859, 861, 871, 897, 900, 921, 930, 949, 951, 975, 981, 996, 999, 1034, 1045, 1050, 1072, 1073, 1081, 1089, 1092, 1102, 1108, 1120, 1128, 1132, 1145, 1163, 1165, 1170, 1227, 1235, 1236, 1258, 1265, 1273, 1280, 1291, 1293, 1297, 1301, 1317, 1338, 1339, 1366, 1369, 1377, 1380, 1387, 1391, 1393, 1395, 1403, 1418, 1422, 1450, 1455, 1461, 1469, 1479, 1486, 1494, 1507, 1526, 1527, 1532, 1537, 1544, 1546, 1563, 1566, 1570, 1576, 1596, 1600, 1603, 1605, 1620, 1626, 1627, 1634, 1640, 1662, 1665, 1670, 1671, 1684, 1692, 1721, 1735, 1746, 1747, 1768, 1785, 1798, 1803, 1825, 1830, 1833, 1847, 1850, 1881, 1892, 1899, 1904, 1923, 1938, 1942, 1943, 1944, 1976, 1988, 1994, 2006, 2022, 2045, 2050, 2057, 2059, 2061, 2064, 2071, 2083, 2087, 2120, 2125, 2132, 2145, 2172, 2203, 2221, 2253, 2258, 2269, 2306, 2311, 2325, 2337, 2346, 2373, 2384, 2424, 2440, 2449, 2457, ...
19 | 2, 10, 11, 12, 14, 19, 24, 40, 45, 46, 48, 65, 66, 67, 75, 85, 90, 103, 105, 117, 119, 137, 147, 164, 167, 179, 181, 205, 220, 235, 242, 253, 254, 263, 268, 277, 303, 315, 332, 337, 366, 369, 370, 389, 399, 404, 424, 431, 446, 449, 480, 481, 506, 509, 521, 523, 531, 547, 567, 573, 581, 622, 646, 651, 673, 736, 761, 768, 787, 797, 807, 810, 811, 817, 840, 846, 857, 867, 869, 870, 888, 899, 902, 951, 971, 988, 990, 992, 1010, 1023, 1027, 1030, 1041, 1042, 1072, 1102, 1112, 1116, 1143, 1147, 1208, 1218, 1223, 1252, 1256, 1257, 1260, 1291, 1330, 1335, 1340, 1402, 1454, 1502, 1536, 1538, 1594, 1622, 1625, 1631, 1642, 1648, 1693, 1701, 1709, 1726, 1734, 1759, 1767, 1779, 1794, 1809, 1822, 1858, 1939, 1954, 1977, 1988, 1992, 1999, 2024, 2030, 2065, 2067, 2081, 2101, 2135, 2150, 2180, 2193, 2196, 2219, 2239, 2293, 2325, 2330, 2336, 2338, 2358, 2382, 2390, 2402, 2403, 2409, 2457, 2461, 2478, 2480, 2499, ...
20 | 2, 3, 4, 9, 11, 12, 16, 19, 26, 34, 45, 48, 54, 63, 70, 78, 82, 83, 86, 91, 96, 97, 101, 105, 109, 110, 119, 120, 126, 129, 139, 141, 149, 153, 167, 168, 171, 173, 181, 182, 187, 190, 192, 195, 215, 227, 228, 229, 257, 260, 276, 298, 299, 305, 309, 311, 314, 319, 334, 339, 352, 362, 369, 373, 375, 378, 414, 420, 425, 433, 444, 462, 467, 470, 477, 478, 479, 482, 483, 485, 506, 518, 519, 532, 534, 540, 548, 550, 552, 564, 570, 575, 591, 593, 601, 605, 606, 619, 626, 631, 642, 646, 649, 659, 660, 673, 684, 714, 719, 731, 734, 735, 749, 757, 791, 804, 832, 837, 842, 851, 876, 884, 885, 890, 899, 913, 926, 928, 937, 946, 949, 955, 961, 962, 965, 972, 987, 1010, 1044, 1053, 1057, 1085, 1095, 1098, 1104, 1114, 1118, 1125, 1141, 1162, 1170, 1188, 1201, 1207, 1217, 1218, 1220, 1224, 1229, 1234, 1239, 1262, 1294, 1306, 1322, 1327, 1342, 1349, 1362, 1369, 1384, 1394, 1400, 1401, 1423, 1464, 1469, 1475, 1477, 1490, 1491, 1511, 1513, 1521, 1523, 1531, 1534, 1540, 1544, 1567, 1569, 1571, 1585, 1593, 1611, 1617, 1628, 1630, 1631, 1637, 1640, 1641, 1650, 1665, 1672, 1674, 1677, 1699, 1705, 1706, 1710, 1711, 1719, 1721, 1739, 1744, 1763, 1767, 1770, 1794, 1810, 1815, 1817, 1818, 1820, 1844, 1869, 1879, 1886, 1892, 1898, 1902, 1918, 1920, 1930, 1931, 1937, 1942, 1945, 1961, 1964, 1967, 1971, 1978, 1999, 2002, 2016, 2031, 2038, 2043, 2057, 2062, 2068, 2073, 2075, 2079, 2114, 2116, 2117, 2155, 2174, 2185, 2189, 2191, 2205, 2215, 2221, 2226, 2236, 2237, 2239, 2248, 2259, 2269, 2270, 2306, 2307, 2313, 2315, 2338, 2360, 2372, 2377, 2393, 2395, 2409, 2418, 2422, 2429, 2430, 2437, 2474, 2482, 2487, ...
21 | 2, 3, 6, 7, 12, 18, 22, 27, 28, 30, 35, 41, 59, 63, 69, 72, 112, 116, 127, 128, 132, 133, 136, 140, 156, 164, 166, 172, 191, 198, 202, 215, 218, 219, 247, 276, 288, 305, 307, 323, 334, 343, 377, 383, 433, 474, 479, 516, 519, 521, 532, 534, 538, 549, 575, 586, 613, 622, 646, 647, 675, 680, 688, 692, 733, 737, 790, 793, 795, 815, 822, 902, 909, 911, 915, 947, 952, 966, 996, 1025, 1033, 1034, 1048, 1073, 1075, 1082, 1087, 1093, 1129, 1144, 1163, 1166, 1182, 1188, 1208, 1241, 1251, 1252, 1269, 1274, 1310, 1340, 1363, 1379, 1418, 1424, 1460, 1461, 1463, 1494, 1501, 1503, 1507, 1508, 1510, 1532, 1540, 1553, 1568, 1590, 1595, 1603, 1623, 1627, 1635, 1660, 1682, 1719, 1728, 1749, 1756, 1766, 1790, 1796, 1797, 1826, 1832, 1857, 1879, 1886, 1924, 1929, 1939, 1954, 1974, 1979, 1982, 1983, 1997, 2004, 2012, 2013, 2064, 2075, 2090, 2097, 2099, 2107, 2113, 2118, 2123, 2128, 2134, 2152, 2200, 2205, 2221, 2234, 2243, 2252, 2257, 2262, 2280, 2306, 2320, 2322, 2323, 2376, 2381, 2383, 2410, 2424, 2436, 2450, 2459, 2480, 2483, ...
22 | 2, 6, 12, 13, 23, 24, 26, 35, 54, 62, 69, 91, 105, 147, 160, 163, 183, 185, 193, 229, 232, 233, 236, 248, 252, 262, 269, 280, 285, 294, 303, 307, 315, 330, 376, 394, 426, 430, 439, 440, 464, 469, 491, 492, 508, 537, 625, 629, 647, 653, 666, 752, 772, 775, 786, 788, 832, 840, 852, 854, 855, 859, 905, 922, 972, 993, 998, 1012, 1041, 1043, 1061, 1076, 1082, 1129, 1136, 1154, 1158, 1173, 1177, 1199, 1208, 1220, 1237, 1266, 1277, 1312, 1336, 1338, 1370, 1386, 1405, 1429, 1439, 1442, 1451, 1455, 1508, 1522, 1543, 1553, 1564, 1566, 1626, 1665, 1734, 1775, 1810, 1821, 1829, 1840, 1844, 1908, 1910, 1938, 1963, 1982, 2009, 2017, 2040, 2070, 2083, 2101, 2105, 2118, 2155, 2163, 2170, 2201, 2203, 2224, 2231, 2244, 2329, 2346, 2347, 2373, 2433, 2450, 2456, 2461, 2485, 2486, 2493, 2496, ...
23 | 10, 40, 82, 113, 127, 141, 170, 257, 275, 287, 295, 315, 344, 373, 442, 468, 599, 609, 634, 646, 663, 671, 710, 819, 834, 857, 884, 894, 904, 992, 997, 1060, 1069, 1077, 1120, 1143, 1190, 1232, 1253, 1261, 1280, 1291, 1347, 1407, 1436, 1448, 1483, 1514, 1570, 1642, 1680, 1684, 1691, 1744, 1749, 1786, 1801, 1816, 1830, 1879, 1891, 1906, 1946, 1968, 1994, 2005, 2158, 2200, 2318, 2444, 2459, ...
24 | 2, 3, 5, 6, 10, 13, 14, 15, 18, 25, 26, 27, 31, 34, 37, 39, 40, 42, 44, 46, 49, 50, 53, 59, 62, 63, 65, 68, 69, 75, 76, 77, 83, 87, 99, 100, 102, 104, 107, 110, 142, 152, 161, 166, 174, 175, 184, 187, 188, 191, 192, 199, 204, 207, 208, 213, 215, 222, 224, 227, 238, 244, 245, 247, 250, 256, 259, 263, 264, 265, 268, 274, 279, 286, 289, 303, 310, 311, 312, 314, 327, 332, 337, 339, 350, 353, 363, 366, 376, 380, 396, 409, 411, 434, 438, 439, 451, 466, 467, 486, 499, 501, 503, 506, 508, 520, 549, 550, 551, 555, 575, 579, 581, 588, 593, 611, 612, 615, 616, 619, 623, 630, 639, 643, 645, 646, 649, 653, 658, 661, 663, 672, 673, 679, 680, 690, 694, 714, 716, 718, 730, 731, 741, 743, 753, 759, 761, 764, 781, 792, 799, 813, 816, 818, 834, 837, 842, 854, 861, 863, 867, 872, 873, 875, 878, 880, 886, 892, 903, 937, 965, 978, 989, 990, 1034, 1036, 1041, 1051, 1055, 1080, 1081, 1082, 1095, 1096, 1104, 1105, 1108, 1122, 1127, 1129, 1133, 1135, 1136, 1148, 1171, 1193, 1217, 1221, 1222, 1241, 1242, 1247, 1250, 1253, 1254, 1266, 1280, 1281, 1283, 1287, 1292, 1298, 1302, 1308, 1314, 1323, 1330, 1332, 1340, 1355, 1359, 1361, 1373, 1375, 1378, 1379, 1381, 1384, 1389, 1398, 1399, 1411, 1413, 1424, 1445, 1449, 1457, 1461, 1462, 1473, 1483, 1484, 1486, 1487, 1501, 1505, 1521, 1523, 1544, 1557, 1562, 1573, 1583, 1584, 1586, 1590, 1591, 1597, 1610, 1615, 1630, 1638, 1644, 1651, 1652, 1654, 1660, 1664, 1670, 1681, 1688, 1690, 1699, 1704, 1705, 1706, 1707, 1708, 1713, 1715, 1723, 1741, 1743, 1752, 1757, 1767, 1771, 1772, 1775, 1778, 1780, 1797, 1798, 1816, 1822, 1829, 1838, 1848, 1853, 1859, 1869, 1871, 1893, 1904, 1934, 1952, 1957, 1971, 1972, 1973, 1974, 1990, 1998, 2003, 2010, 2011, 2018, 2020, 2022, 2026, 2030, 2031, 2040, 2050, 2070, 2078, 2090, 2094, 2102, 2108, 2115, 2123, 2128, 2129, 2142, 2150, 2156, 2192, 2200, 2208, 2213, 2218, 2238, 2239, 2250, 2264, 2268, 2279, 2295, 2301, 2307, 2310, 2314, 2325, 2334, 2335, 2340, 2346, 2348, 2354, 2364, 2376, 2380, 2381, 2387, 2389, 2393, 2400, 2406, 2407, 2415, 2425, 2427, 2444, 2445, 2453, 2462, 2467, 2474, 2487, ...
25 | 22, 33, 39, 43, 46, 62, 74, 101, 134, 141, 142, 151, 167, 212, 238, 266, 287, 313, 335, 369, 414, 415, 418, 432, 436, 509, 516, 541, 604, 679, 697, 770, 782, 815, 831, 859, 871, 874, 895, 897, 924, 1039, 1048, 1070, 1085, 1096, 1111, 1121, 1134, 1145, 1170, 1177, 1212, 1239, 1240, 1262, 1321, 1339, 1347, 1361, 1364, 1374, 1407, 1413, 1414, 1449, 1473, 1488, 1513, 1537, 1544, 1624, 1627, 1628, 1630, 1665, 1682, 1691, 1718, 1744, 1758, 1766, 1783, 1792, 1822, 1835, 1979, 2026, 2043, 2046, 2050, 2118, 2159, 2160, 2161, 2165, 2180, 2214, 2223, 2235, 2248, 2250, 2256, 2296, 2308, 2312, 2357, 2368, 2400, 2414, 2433, 2444, 2450, 2487, 2496, ...
26 | 2, 3, 21, 22, 23, 35, 39, 74, 77, 80, 84, 89, 103, 108, 114, 121, 126, 134, 152, 153, 171, 180, 195, 204, 230, 246, 256, 263, 285, 297, 321, 326, 336, 337, 342, 345, 351, 368, 390, 393, 397, 398, 399, 413, 427, 439, 490, 493, 525, 558, 563, 566, 574, 591, 602, 609, 630, 641, 652, 657, 660, 667, 682, 685, 694, 708, 712, 716, 721, 755, 760, 763, 791, 792, 797, 813, 828, 839, 850, 862, 882, 898, 999, 1025, 1029, 1030, 1035, 1038, 1049, 1063, 1073, 1076, 1081, 1105, 1111, 1128, 1162, 1176, 1178, 1185, 1216, 1227, 1234, 1238, 1264, 1268, 1274, 1307, 1318, 1324, 1359, 1364, 1369, 1371, 1373, 1377, 1388, 1392, 1445, 1450, 1451, 1461, 1489, 1496, 1529, 1539, 1550, 1555, 1560, 1610, 1612, 1616, 1624, 1632, 1689, 1695, 1794, 1805, 1824, 1857, 1873, 1889, 1899, 1930, 1943, 1950, 1955, 1963, 1979, 1985, 1994, 2003, 2005, 2010, 2085, 2103, 2108, 2113, 2122, 2151, 2153, 2172, 2189, 2214, 2228, 2234, 2236, 2257, 2268, 2293, 2307, 2314, 2337, 2373, 2380, 2400, 2417, 2433, 2436, 2443, 2483, 2487, 2492, ...
27 | 2, 11, 44, 45, 56, 62, 63, 67, 110, 127, 163, 170, 208, 214, 219, 232, 234, 245, 261, 263, 325, 333, 337, 394, 395, 398, 402, 413, 421, 428, 434, 437, 498, 502, 505, 557, 558, 578, 610, 633, 637, 643, 652, 691, 692, 695, 715, 723, 731, 750, 761, 774, 794, 797, 804, 806, 826, 846, 854, 855, 859, 863, 906, 919, 923, 926, 943, 977, 982, 1045, 1046, 1066, 1081, 1085, 1086, 1091, 1094, 1108, 1157, 1164, 1181, 1184, 1197, 1207, 1212, 1217, 1239, 1271, 1282, 1298, 1300, 1303, 1341, 1371, 1378, 1402, 1407, 1458, 1525, 1579, 1596, 1624, 1632, 1679, 1696, 1705, 1725, 1772, 1775, 1804, 1809, 1827, 1828, 1845, 1853, 1867, 1896, 1909, 1912, 1914, 1958, 1960, 1961, 2004, 2005, 2007, 2030, 2061, 2089, 2122, 2131, 2137, 2138, 2145, 2167, 2276, 2316, 2319, 2321, 2322, 2332, 2393, 2394, 2404, 2439, 2458, ...
28 | 4, 5, 7, 13, 25, 33, 41, 63, 74, 80, 88, 94, 96, 104, 116, 144, 149, 151, 154, 165, 167, 174, 183, 191, 197, 208, 231, 241, 262, 268, 270, 290, 318, 319, 361, 368, 376, 390, 394, 412, 431, 434, 442, 464, 489, 492, 521, 529, 556, 568, 574, 575, 585, 589, 613, 629, 639, 654, 666, 667, 672, 683, 684, 709, 724, 732, 763, 782, 787, 803, 811, 817, 821, 828, 842, 845, 850, 858, 876, 928, 933, 951, 977, 979, 982, 993, 995, 1006, 1020, 1037, 1045, 1056, 1082, 1086, 1089, 1101, 1111, 1135, 1140, 1156, 1166, 1196, 1217, 1218, 1223, 1224, 1246, 1280, 1285, 1306, 1343, 1357, 1368, 1414, 1426, 1428, 1480, 1484, 1509, 1517, 1530, 1536, 1541, 1567, 1571, 1579, 1589, 1594, 1611, 1707, 1712, 1736, 1739, 1744, 1757, 1760, 1763, 1770, 1781, 1791, 1802, 1807, 1818, 1844, 1855, 1862, 1891, 1930, 1942, 1960, 1972, 1985, 1996, 2029, 2043, 2060, 2076, 2170, 2180, 2198, 2213, 2217, 2250, 2261, 2271, 2329, 2340, 2343, 2377, 2412, 2423, 2431, 2464, 2465, 2470, 2482, ...
29 | 6, 40, 65, 70, 114, 151, 221, 229, 268, 283, 398, 451, 460, 519, 554, 587, 627, 628, 659, 687, 699, 859, 884, 915, 943, 974, 986, 1101, 1120, 1159, 1176, 1212, 1223, 1297, 1312, 1322, 1337, 1390, 1409, 1415, 1446, 1477, 1508, 1592, 1625, 1636, 1683, 1691, 1800, 1802, 1803, 1820, 2020, 2126, 2414, 2421, 2429, 2461, ...
30 | 2, 6, 7, 8, 9, 16, 18, 20, 26, 29, 31, 32, 33, 40, 41, 47, 57, 76, 82, 87, 88, 93, 101, 109, 120, 121, 125, 133, 140, 144, 162, 163, 175, 178, 183, 186, 191, 196, 215, 216, 218, 227, 233, 242, 253, 266, 267, 273, 276, 278, 304, 312, 317, 319, 328, 336, 374, 380, 391, 408, 413, 421, 436, 438, 439, 448, 453, 457, 474, 475, 479, 491, 500, 502, 505, 510, 511, 521, 526, 541, 558, 560, 564, 609, 621, 625, 628, 636, 648, 657, 660, 678, 682, 683, 692, 705, 712, 714, 727, 738, 746, 759, 767, 782, 783, 802, 803, 826, 831, 833, 836, 837, 844, 846, 851, 863, 882, 898, 924, 925, 931, 938, 946, 953, 970, 975, 976, 984, 988, 990, 1011, 1017, 1054, 1091, 1093, 1115, 1118, 1130, 1142, 1145, 1161, 1162, 1182, 1187, 1201, 1219, 1227, 1250, 1260, 1278, 1281, 1294, 1338, 1356, 1364, 1370, 1379, 1389, 1396, 1397, 1402, 1409, 1420, 1433, 1441, 1451, 1459, 1461, 1467, 1486, 1488, 1489, 1492, 1511, 1517, 1535, 1537, 1555, 1565, 1595, 1610, 1614, 1640, 1643, 1644, 1673, 1704, 1709, 1710, 1715, 1724, 1731, 1732, 1735, 1751, 1776, 1786, 1794, 1797, 1800, 1812, 1824, 1827, 1833, 1836, 1844, 1847, 1848, 1859, 1861, 1862, 1865, 1891, 1892, 1917, 1920, 1929, 1941, 1950, 1951, 1983, 1984, 1994, 2011, 2012, 2033, 2045, 2046, 2056, 2074, 2082, 2097, 2104, 2105, 2108, 2118, 2135, 2148, 2149, 2172, 2178, 2196, 2199, 2202, 2203, 2205, 2206, 2207, 2251, 2258, 2264, 2272, 2278, 2294, 2304, 2332, 2360, 2365, 2385, 2388, 2394, 2403, 2434, 2446, 2463, 2465, 2467, 2472, 2488, 2494, ...
31 | 2, 14, 19, 31, 44, 53, 71, 82, 117, 127, 131, 145, 156, 177, 197, 203, 241, 258, 261, 276, 283, 293, 320, 325, 373, 379, 387, 388, 406, 413, 461, 462, 470, 486, 491, 534, 549, 569, 582, 612, 618, 639, 696, 706, 723, 746, 765, 767, 774, 796, 802, 877, 878, 903, 923, 931, 981, 991, 998, 1002, 1008, 1049, 1077, 1113, 1114, 1115, 1138, 1146, 1159, 1177, 1189, 1200, 1229, 1270, 1281, 1312, 1359, 1429, 1486, 1493, 1496, 1558, 1605, 1614, 1626, 1632, 1649, 1673, 1698, 1754, 1790, 1859, 1865, 1894, 1900, 1904, 1918, 1939, 1948, 1973, 1998, 1999, 2022, 2031, 2038, 2055, 2111, 2167, 2169, 2180, 2236, 2300, 2377, 2387, 2388, 2422, 2462, 2473, ...
32 | 2, 3, 9, 29, 41, 44, 73, 74, 76, 81, 87, 94, 111, 113, 156, 157, 158, 167, 173, 176, 187, 188, 195, 198, 199, 248, 253, 288, 295, 301, 306, 309, 318, 330, 348, 370, 371, 382, 391, 396, 403, 435, 441, 452, 456, 470, 474, 476, 478, 485, 525, 560, 568, 575, 585, 589, 598, 599, 607, 617, 642, 657, 669, 686, 688, 690, 736, 774, 776, 778, 779, 789, 790, 830, 832, 834, 846, 900, 905, 916, 946, 955, 956, 969, 972, 982, 984, 995, 1001, 1018, 1023, 1044, 1078, 1083, 1102, 1141, 1147, 1167, 1170, 1176, 1203, 1213, 1214, 1236, 1247, 1252, 1265, 1322, 1349, 1360, 1407, 1408, 1412, 1426, 1457, 1491, 1528, 1535, 1636, 1660, 1674, 1685, 1734, 1747, 1803, 1812, 1822, 1830, 1834, 1835, 1850, 1856, 1860, 1876, 1890, 1891, 1943, 1950, 1979, 1984, 2006, 2027, 2031, 2072, 2078, 2084, 2120, 2135, 2152, 2156, 2159, 2208, 2218, 2230, 2236, 2237, 2271, 2284, 2315, 2318, 2352, 2355, 2377, 2392, 2407, 2415, 2438, 2441, 2448, 2462, ...
33 | 2, 3, 13, 14, 18, 31, 44, 59, 62, 75, 80, 104, 109, 145, 185, 213, 273, 282, 309, 321, 337, 379, 399, 405, 411, 430, 452, 464, 470, 522, 535, 560, 566, 586, 593, 597, 653, 654, 688, 702, 704, 727, 728, 744, 746, 780, 805, 806, 816, 822, 829, 846, 856, 871, 884, 979, 986, 1001, 1014, 1043, 1087, 1102, 1109, 1133, 1185, 1192, 1211, 1324, 1325, 1340, 1410, 1422, 1474, 1476, 1494, 1503, 1556, 1636, 1648, 1661, 1672, 1716, 1720, 1721, 1723, 1725, 1806, 1810, 1829, 1927, 1975, 1983, 1989, 2085, 2168, 2174, 2189, 2207, 2210, 2275, 2290, 2328, 2331, 2359, 2360, 2369, 2379, 2397, 2419, 2487, ...
34 | 2, 7, 13, 17, 19, 41, 48, 58, 59, 66, 84, 86, 129, 133, 139, 143, 146, 149, 166, 167, 231, 268, 270, 299, 305, 322, 328, 359, 387, 397, 408, 469, 523, 527, 534, 541, 553, 555, 569, 577, 582, 583, 600, 608, 634, 662, 664, 667, 672, 673, 709, 714, 720, 725, 733, 746, 747, 759, 776, 802, 808, 822, 860, 870, 877, 892, 896, 902, 911, 962, 970, 975, 985, 1034, 1050, 1051, 1070, 1082, 1090, 1109, 1148, 1150, 1168, 1185, 1198, 1200, 1209, 1212, 1233, 1235, 1261, 1281, 1312, 1338, 1339, 1357, 1380, 1382, 1387, 1403, 1411, 1440, 1441, 1459, 1477, 1483, 1497, 1503, 1560, 1561, 1578, 1583, 1689, 1691, 1710, 1751, 1752, 1755, 1789, 1797, 1826, 1883, 1886, 1887, 1901, 1912, 1950, 1955, 1965, 1971, 1986, 2016, 2032, 2038, 2041, 2060, 2071, 2074, 2083, 2093, 2121, 2126, 2139, 2180, 2182, 2193, 2195, 2209, 2247, 2260, 2292, 2294, 2340, 2461, 2493, ...
35 | 14, 33, 39, 55, 112, 130, 132, 138, 168, 176, 179, 186, 189, 195, 210, 246, 259, 264, 310, 318, 346, 417, 431, 467, 478, 480, 534, 545, 564, 567, 661, 671, 741, 744, 749, 757, 786, 794, 804, 825, 851, 866, 911, 948, 955, 962, 976, 992, 1014, 1033, 1042, 1082, 1109, 1161, 1193, 1220, 1244, 1260, 1268, 1278, 1313, 1414, 1437, 1476, 1563, 1615, 1618, 1658, 1687, 1715, 1738, 1741, 1829, 2016, 2019, 2044, 2045, 2085, 2103, 2105, 2110, 2114, 2118, 2150, 2181, 2198, 2215, 2248, 2260, 2262, 2277, 2286, 2328, 2343, 2366, 2394, 2415, 2440, 2445, 2468, ...
36 | 3, 7, 9, 10, 11, 14, 27, 30, 33, 37, 40, 51, 62, 71, 78, 80, 82, 95, 117, 125, 137, 142, 148, 157, 185, 189, 191, 197, 223, 232, 236, 252, 255, 265, 287, 321, 323, 324, 326, 332, 337, 341, 354, 369, 391, 414, 416, 418, 430, 487, 509, 545, 558, 563, 582, 588, 589, 618, 675, 694, 714, 715, 739, 773, 776, 788, 884, 898, 909, 928, 932, 935, 948, 950, 952, 969, 989, 996, 1008, 1029, 1040, 1042, 1079, 1085, 1094, 1119, 1124, 1136, 1148, 1159, 1183, 1196, 1207, 1211, 1281, 1296, 1377, 1379, 1380, 1474, 1486, 1516, 1517, 1526, 1538, 1554, 1562, 1563, 1566, 1579, 1582, 1599, 1619, 1628, 1649, 1653, 1655, 1657, 1671, 1773, 1792, 1804, 1809, 1822, 1841, 1849, 1864, 1896, 1912, 1913, 1991, 2012, 2046, 2061, 2071, 2112, 2113, 2115, 2143, 2147, 2160, 2180, 2186, 2211, 2248, 2249, 2263, 2268, 2283, 2302, 2315, 2322, 2347, 2372, 2399, 2426, 2436, 2443, ...
37 | 61, 77, 94, 97, 99, 113, 126, 130, 134, 147, 161, 172, 187, 202, 208, 246, 261, 273, 285, 302, 320, 432, 444, 503, 523, 525, 563, 666, 680, 709, 740, 757, 787, 902, 962, 964, 969, 1013, 1109, 1111, 1165, 1186, 1219, 1351, 1528, 1587, 1660, 1698, 1710, 1754, 1811, 1972, 2070, 2088, 2137, 2145, 2185, 2210, 2295, 2305, 2348, 2365, 2396, 2398, 2439, 2453, 2462, ...
38 | 2, 10, 13, 18, 24, 28, 43, 47, 49, 54, 65, 83, 98, 106, 143, 152, 177, 184, 190, 194, 195, 208, 241, 249, 259, 264, 292, 315, 319, 345, 353, 355, 386, 394, 481, 500, 517, 525, 534, 535, 550, 556, 595, 601, 649, 656, 680, 686, 687, 697, 707, 710, 756, 798, 804, 817, 818, 829, 839, 841, 858, 864, 891, 906, 912, 932, 948, 973, 991, 994, 1012, 1030, 1032, 1036, 1046, 1057, 1060, 1066, 1071, 1089, 1166, 1175, 1203, 1205, 1227, 1234, 1237, 1261, 1283, 1284, 1319, 1327, 1335, 1336, 1337, 1356, 1372, 1387, 1412, 1413, 1439, 1450, 1460, 1483, 1488, 1495, 1502, 1515, 1518, 1521, 1532, 1538, 1676, 1677, 1685, 1686, 1690, 1720, 1722, 1734, 1739, 1758, 1759, 1762, 1782, 1792, 1814, 1822, 1833, 1834, 1838, 1839, 1842, 1861, 1869, 1884, 1918, 1922, 1972, 1975, 1999, 2005, 2012, 2033, 2042, 2056, 2093, 2119, 2221, 2244, 2251, 2264, 2325, 2356, 2359, 2371, 2381, 2383, 2427, 2436, 2446, 2479, ...
39 | 10, 23, 29, 86, 240, 254, 284, 308, 340, 369, 371, 378, 382, 393, 407, 422, 448, 459, 574, 582, 613, 619, 667, 686, 703, 767, 769, 844, 851, 861, 875, 881, 944, 987, 995, 1030, 1056, 1207, 1219, 1233, 1251, 1270, 1279, 1292, 1343, 1372, 1399, 1409, 1445, 1468, 1497, 1500, 1557, 1586, 1598, 1628, 1633, 1645, 1667, 1677, 1712, 1760, 1803, 1807, 1835, 1933, 2004, 2014, 2028, 2039, 2057, 2113, 2136, 2143, 2196, 2224, 2229, 2271, 2291, 2376, 2391, 2397, 2398, ...
40 | 2, 3, 4, 31, 37, 39, 46, 77, 98, 119, 124, 143, 144, 154, 169, 197, 205, 232, 245, 266, 291, 295, 297, 305, 308, 319, 332, 333, 413, 426, 431, 437, 454, 459, 472, 483, 513, 528, 531, 542, 579, 617, 619, 635, 647, 677, 724, 737, 748, 780, 806, 815, 819, 820, 840, 851, 858, 870, 875, 907, 920, 927, 940, 983, 1002, 1028, 1048, 1050, 1064, 1109, 1125, 1149, 1179, 1197, 1209, 1222, 1229, 1233, 1244, 1248, 1251, 1320, 1337, 1345, 1410, 1434, 1471, 1522, 1526, 1566, 1574, 1576, 1578, 1607, 1723, 1725, 1738, 1758, 1794, 1877, 1885, 1886, 1968, 1999, 2000, 2009, 2010, 2023, 2041, 2058, 2064, 2073, 2093, 2136, 2141, 2148, 2171, 2182, 2183, 2204, 2215, 2222, 2232, 2273, 2275, 2287, 2317, 2323, 2342, 2355, 2364, 2396, 2423, 2428, 2450, 2478, 2487, ...
41 | 14, 53, 55, 58, 71, 76, 82, 211, 248, 271, 296, 316, 430, 433, 439, 472, 545, 553, 555, 596, 663, 677, 682, 746, 814, 832, 885, 926, 947, 959, 1048, 1121, 1146, 1175, 1207, 1300, 1302, 1390, 1404, 1445, 1466, 1477, 1541, 1650, 1699, 1702, 1706, 1756, 1757, 1782, 1902, 1944, 1963, 1994, 2012, 2042, 2122, 2163, 2215, 2344, 2348, 2369, 2439, 2442, 2465, ...
42 | 2, 6, 8, 11, 17, 23, 35, 41, 49, 58, 65, 85, 88, 96, 97, 103, 107, 111, 129, 139, 144, 161, 170, 194, 197, 213, 214, 217, 223, 230, 236, 238, 247, 274, 283, 290, 297, 298, 301, 323, 339, 348, 360, 381, 382, 383, 389, 393, 398, 403, 427, 445, 452, 473, 480, 497, 500, 511, 523, 561, 575, 595, 601, 604, 606, 615, 629, 651, 652, 680, 685, 702, 705, 725, 762, 770, 774, 780, 781, 782, 805, 814, 857, 892, 912, 918, 942, 987, 1004, 1027, 1030, 1039, 1045, 1054, 1099, 1125, 1126, 1131, 1155, 1160, 1168, 1169, 1215, 1231, 1235, 1246, 1249, 1256, 1271, 1279, 1289, 1291, 1314, 1315, 1325, 1373, 1383, 1385, 1386, 1415, 1416, 1420, 1436, 1454, 1460, 1468, 1476, 1498, 1519, 1522, 1556, 1585, 1629, 1657, 1665, 1687, 1692, 1693, 1708, 1713, 1715, 1716, 1729, 1730, 1735, 1744, 1779, 1784, 1787, 1794, 1801, 1808, 1810, 1812, 1816, 1842, 1847, 1851, 1857, 1863, 1914, 1924, 1958, 1966, 1974, 1975, 1991, 2001, 2002, 2013, 2022, 2029, 2036, 2052, 2124, 2132, 2138, 2157, 2182, 2207, 2218, 2228, 2229, 2246, 2251, 2257, 2276, 2279, 2280, 2295, 2303, 2338, 2344, 2360, 2362, 2378, 2387, 2389, 2396, 2406, 2408, 2424, 2465, 2472, 2473, 2486, ...
43 | 15, 21, 26, 86, 89, 114, 123, 163, 180, 310, 332, 377, 409, 438, 448, 457, 477, 526, 534, 556, 586, 612, 653, 665, 690, 692, 709, 760, 783, 803, 821, 848, 877, 899, 909, 942, 981, 1041, 1042, 1043, 1066, 1068, 1069, 1106, 1126, 1139, 1197, 1310, 1386, 1476, 1561, 1599, 1606, 1636, 1774, 1802, 1835, 1867, 2033, 2068, 2153, 2155, 2229, 2237, 2256, 2269, 2276, 2324, 2467, ...
44 | 25, 60, 70, 77, 86, 116, 117, 119, 137, 151, 166, 175, 176, 177, 181, 184, 202, 208, 229, 241, 300, 313, 328, 332, 344, 385, 402, 442, 444, 446, 453, 467, 470, 500, 501, 522, 527, 531, 532, 534, 568, 584, 592, 629, 690, 696, 701, 706, 709, 728, 734, 749, 795, 812, 828, 838, 844, 845, 855, 857, 950, 951, 965, 991, 1009, 1011, 1084, 1090, 1098, 1102, 1125, 1142, 1168, 1169, 1170, 1179, 1181, 1191, 1212, 1213, 1234, 1246, 1247, 1262, 1284, 1285, 1304, 1310, 1311, 1358, 1368, 1387, 1411, 1421, 1432, 1459, 1468, 1483, 1485, 1515, 1538, 1542, 1570, 1572, 1587, 1616, 1618, 1628, 1629, 1636, 1646, 1647, 1654, 1662, 1676, 1687, 1710, 1723, 1732, 1734, 1750, 1804, 1807, 1831, 1861, 1865, 1875, 1882, 1899, 1921, 1925, 1942, 1945, 1996, 2014, 2015, 2023, 2036, 2066, 2071, 2078, 2088, 2136, 2149, 2165, 2223, 2232, 2262, 2281, 2287, 2435, 2440, 2462, 2463, 2498, ...
45 | 11, 18, 23, 24, 29, 33, 40, 51, 61, 65, 85, 91, 95, 113, 153, 206, 207, 218, 264, 276, 285, 292, 313, 332, 347, 362, 367, 391, 417, 418, 424, 426, 428, 439, 444, 451, 454, 468, 476, 493, 542, 601, 634, 642, 653, 657, 680, 705, 734, 764, 783, 787, 856, 867, 873, 915, 942, 945, 953, 964, 971, 1013, 1022, 1044, 1054, 1058, 1070, 1111, 1133, 1152, 1242, 1252, 1274, 1304, 1322, 1334, 1365, 1404, 1405, 1406, 1423, 1438, 1442, 1456, 1483, 1500, 1524, 1527, 1531, 1544, 1553, 1562, 1582, 1662, 1671, 1688, 1723, 1736, 1743, 1751, 1761, 1812, 1830, 1833, 1835, 1837, 1910, 1946, 1985, 1989, 2017, 2022, 2026, 2091, 2095, 2129, 2143, 2166, 2170, 2215, 2271, 2272, 2282, 2283, 2288, 2297, 2300, 2305, 2306, 2308, 2329, 2336, 2364, 2389, 2402, 2416, 2418, 2443, 2448, 2452, 2466, 2472, 2485, 2496, ...
46 | 2, 3, 7, 16, 17, 18, 25, 46, 47, 106, 110, 111, 118, 136, 144, 145, 230, 238, 361, 382, 422, 439, 474, 494, 495, 519, 588, 639, 657, 707, 733, 751, 802, 831, 863, 902, 925, 976, 994, 1001, 1059, 1140, 1181, 1240, 1249, 1319, 1375, 1442, 1458, 1508, 1699, 1724, 1734, 1751, 1757, 1760, 1766, 1807, 1849, 1897, 1904, 1914, 1922, 1931, 1978, 2029, 2057, 2075, 2139, 2166, 2183, 2209, 2216, 2230, 2241, 2246, 2270, 2307, 2309, 2367, 2429, 2446, 2456, 2460, ...
47 | 5, 17, 19, 55, 62, 75, 89, 98, 99, 132, 172, 186, 197, 220, 268, 278, 279, 288, 439, 443, 496, 579, 583, 587, 742, 777, 825, 911, 966, 1003, 1029, 1155, 1202, 1267, 1377, 1461, 1472, 1473, 1530, 1562, 1634, 1682, 1705, 1783, 1848, 1852, 1885, 1900, 1943, 1957, 1969, 2036, 2044, 2105, 2111, 2265, 2282, 2485, 2492, ...
48 | 5, 7, 10, 16, 17, 39, 41, 51, 55, 70, 96, 97, 104, 132, 136, 137, 148, 182, 195, 206, 216, 227, 234, 236, 245, 262, 267, 278, 283, 287, 305, 310, 331, 361, 362, 380, 384, 409, 410, 411, 423, 438, 446, 464, 473, 495, 522, 549, 561, 564, 578, 587, 591, 592, 602, 619, 622, 633, 665, 679, 687, 714, 741, 759, 777, 784, 813, 828, 830, 847, 849, 853, 874, 879, 924, 931, 940, 977, 1015, 1030, 1037, 1057, 1058, 1090, 1107, 1141, 1145, 1160, 1170, 1211, 1246, 1265, 1285, 1296, 1299, 1304, 1306, 1312, 1313, 1316, 1328, 1373, 1393, 1400, 1409, 1420, 1443, 1500, 1518, 1542, 1546, 1557, 1570, 1574, 1643, 1644, 1653, 1656, 1685, 1686, 1688, 1727, 1737, 1756, 1759, 1761, 1762, 1763, 1780, 1792, 1807, 1900, 1924, 1927, 1930, 1936, 1959, 1964, 1977, 2015, 2018, 2019, 2033, 2041, 2050, 2059, 2070, 2089, 2122, 2133, 2179, 2208, 2217, 2222, 2255, 2302, 2313, 2318, 2342, 2350, 2384, 2407, 2416, 2417, 2430, 2440, 2443, 2470, 2477, 2487, ...
49 | 2, 5, 50, 57, 62, 67, 104, 123, 138, 151, 155, 186, 194, 233, 236, 253, 261, 269, 306, 337, 359, 416, 425, 439, 557, 613, 695, 705, 742, 748, 757, 821, 832, 884, 885, 892, 897, 902, 936, 937, 988, 1125, 1165, 1176, 1248, 1265, 1320, 1337, 1386, 1401, 1544, 1590, 1594, 1675, 1707, 1712, 1714, 1722, 1725, 1798, 1840, 1844, 2037, 2106, 2117, 2148, 2153, 2154, 2155, 2175, 2192, 2208, 2235, 2286, 2292, 2376, 2384, 2416, 2422, 2470, ...
50 | 6, 7, 17, 29, 42, 44, 56, 59, 62, 106, 108, 112, 141, 161, 163, 173, 194, 210, 212, 229, 231, 254, 296, 306, 342, 360, 382, 384, 395, 400, 416, 418, 431, 435, 457, 458, 484, 492, 517, 519, 530, 531, 540, 551, 579, 609, 618, 637, 644, 672, 697, 713, 722, 724, 732, 790, 791, 794, 810, 839, 850, 852, 876, 898, 902, 914, 926, 928, 937, 938, 970, 1010, 1091, 1118, 1123, 1129, 1150, 1157, 1165, 1182, 1214, 1229, 1247, 1253, 1258, 1269, 1272, 1320, 1321, 1378, 1399, 1402, 1422, 1425, 1474, 1500, 1532, 1539, 1573, 1578, 1584, 1586, 1588, 1622, 1624, 1635, 1651, 1653, 1696, 1699, 1725, 1756, 1811, 1875, 1912, 1948, 1970, 2017, 2076, 2092, 2114, 2115, 2119, 2124, 2131, 2133, 2147, 2175, 2219, 2253, 2260, 2295, 2347, 2406, 2462, 2463, 2464, ...
51 | 30, 39, 42, 56, 73, 99, 114, 142, 193, 214, 263, 268, 320, 329, 381, 397, 432, 435, 446, 476, 477, 499, 538, 580, 584, 585, 652, 685, 689, 837, 845, 867, 954, 1040, 1108, 1160, 1164, 1283, 1302, 1441, 1450, 1456, 1469, 1609, 1645, 1718, 1720, 1759, 2004, 2065, 2068, 2206, 2289, 2332, 2377, 2383, 2416, 2438, 2443, 2458, ...
52 | 11, 16, 37, 70, 96, 121, 131, 187, 280, 281, 317, 396, 441, 501, 554, 582, 696, 704, 733, 735, 765, 831, 876, 891, 894, 967, 1013, 1014, 1049, 1071, 1085, 1096, 1119, 1129, 1151, 1162, 1170, 1281, 1289, 1435, 1544, 1594, 1597, 1620, 1653, 1697, 1738, 1740, 1774, 1831, 1864, 1893, 1895, 1899, 1924, 1931, 1944, 1948, 1955, 1977, 2007, 2014, 2021, 2083, 2084, 2186, 2215, 2216, 2268, 2283, 2375, 2422, 2461, ...
53 | 24, 45, 60, 165, 235, 272, 285, 298, 307, 381, 416, 429, 623, 799, 858, 924, 929, 936, 1034, 1067, 1076, 1130, 1223, 1243, 1327, 1488, 1592, 1594, 1659, 1673, 1676, 1678, 1762, 1873, 1917, 1928, 1997, 2110, 2117, 2141, 2166, 2231, 2243, 2293, 2338, 2380, 2420, 2424, 2428, 2463, 2466, ...
54 | 2, 7, 9, 15, 23, 24, 25, 42, 62, 63, 78, 91, 122, 151, 159, 201, 208, 222, 248, 252, 277, 281, 284, 286, 291, 299, 316, 340, 342, 378, 395, 440, 445, 460, 478, 498, 522, 528, 532, 558, 580, 600, 668, 688, 701, 710, 726, 743, 752, 757, 763, 766, 771, 778, 787, 796, 798, 805, 812, 856, 864, 882, 893, 895, 963, 981, 987, 992, 1014, 1031, 1046, 1061, 1088, 1098, 1108, 1128, 1163, 1170, 1182, 1191, 1201, 1234, 1266, 1269, 1343, 1373, 1383, 1386, 1460, 1467, 1487, 1508, 1535, 1544, 1575, 1588, 1598, 1650, 1662, 1668, 1697, 1717, 1730, 1738, 1750, 1762, 1766, 1795, 1847, 1852, 1869, 1878, 1894, 1905, 1910, 1912, 1925, 1932, 1952, 1963, 1996, 2015, 2018, 2021, 2105, 2108, 2122, 2141, 2161, 2170, 2188, 2194, 2203, 2235, 2238, 2245, 2251, 2266, 2335, 2366, 2379, 2403, 2431, 2496, ...
55 | 7, 14, 39, 53, 61, 83, 127, 148, 154, 199, 208, 228, 249, 281, 288, 340, 430, 435, 476, 511, 525, 533, 534, 541, 594, 663, 668, 670, 671, 678, 683, 728, 739, 821, 830, 865, 867, 871, 873, 891, 922, 982, 992, 1037, 1038, 1059, 1090, 1206, 1212, 1252, 1311, 1413, 1443, 1456, 1522, 1537, 1544, 1575, 1616, 1639, 1651, 1668, 1699, 1735, 1741, 1742, 1754, 1756, 1790, 1806, 1820, 1821, 1823, 1958, 2005, 2058, 2064, 2149, 2167, 2169, 2188, 2194, 2226, 2291, 2316, ...
56 | 2, 5, 12, 19, 23, 33, 34, 84, 130, 136, 159, 167, 183, 187, 192, 197, 226, 229, 253, 259, 260, 269, 274, 323, 332, 337, 381, 404, 444, 464, 487, 491, 496, 520, 527, 577, 583, 614, 666, 672, 678, 683, 688, 702, 742, 747, 748, 762, 790, 830, 857, 865, 869, 871, 898, 901, 946, 1016, 1163, 1245, 1273, 1275, 1280, 1361, 1362, 1371, 1386, 1410, 1457, 1473, 1476, 1488, 1515, 1537, 1559, 1563, 1567, 1570, 1574, 1581, 1600, 1614, 1647, 1689, 1699, 1725, 1769, 1787, 1796, 1804, 1809, 1811, 1837, 1861, 1883, 1888, 1917, 1940, 1964, 1986, 2006, 2027, 2039, 2053, 2137, 2141, 2155, 2179, 2200, 2221, 2227, 2232, 2241, 2318, 2339, 2341, 2345, 2359, 2377, 2389, 2391, 2396, 2401, ...
57 | 5, 42, 45, 54, 95, 125, 145, 147, 166, 211, 250, 274, 338, 407, 445, 522, 619, 621, 636, 648, 649, 657, 691, 733, 745, 775, 808, 878, 903, 920, 946, 1040, 1064, 1097, 1105, 1122, 1137, 1167, 1168, 1188, 1190, 1197, 1209, 1210, 1211, 1269, 1280, 1319, 1374, 1443, 1477, 1491, 1514, 1520, 1553, 1565, 1586, 1592, 1596, 1624, 1630, 1639, 1656, 1667, 1703, 1769, 1775, 1778, 1790, 1824, 1900, 1918, 1925, 1935, 1939, 1952, 1996, 2023, 2076, 2151, 2168, 2201, 2258, 2294, 2325, 2331, 2333, 2353, 2369, 2407, 2460, 2487, ...
58 | 7, 15, 25, 62, 119, 123, 154, 202, 245, 285, 294, 295, 357, 371, 476, 626, 664, 690, 708, 723, 737, 768, 783, 803, 825, 826, 835, 841, 842, 867, 871, 897, 904, 934, 953, 1066, 1069, 1088, 1097, 1108, 1183, 1197, 1202, 1259, 1302, 1364, 1461, 1478, 1497, 1528, 1559, 1638, 1715, 1760, 1764, 1815, 1821, 1849, 1854, 1874, 2047, 2086, 2091, 2100, 2133, 2160, 2231, 2242, 2316, 2422, ...
59 | 19, 70, 102, 116, 126, 188, 209, 257, 294, 359, 451, 461, 468, 470, 638, 653, 710, 762, 766, 781, 824, 901, 939, 964, 995, 1036, 1047, 1098, 1150, 1211, 1234, 1243, 1256, 1278, 1389, 1401, 1422, 1436, 1454, 1492, 1523, 1639, 1703, 1705, 1826, 1913, 1915, 1978, 2013, 2043, 2131, 2140, 2200, 2208, 2228, 2229, 2307, 2309, 2323, 2365, 2480, 2485, ...
60 | 3, 4, 11, 12, 14, 25, 48, 57, 62, 70, 84, 93, 100, 126, 135, 138, 167, 170, 171, 184, 186, 194, 198, 205, 212, 217, 220, 222, 239, 240, 285, 292, 300, 318, 321, 339, 361, 378, 390, 391, 404, 419, 461, 500, 501, 509, 526, 546, 574, 606, 622, 635, 670, 671, 675, 717, 721, 723, 744, 747, 778, 789, 802, 804, 809, 822, 832, 903, 919, 929, 937, 965, 1001, 1012, 1016, 1024, 1034, 1050, 1084, 1086, 1095, 1101, 1132, 1158, 1164, 1174, 1187, 1211, 1215, 1217, 1223, 1262, 1266, 1276, 1282, 1303, 1320, 1342, 1343, 1346, 1365, 1394, 1395, 1403, 1425, 1430, 1450, 1485, 1502, 1504, 1505, 1506, 1624, 1676, 1705, 1711, 1719, 1722, 1741, 1747, 1748, 1750, 1753, 1766, 1797, 1806, 1859, 1870, 1882, 1900, 1912, 1913, 1916, 1925, 1994, 2001, 2004, 2027, 2041, 2085, 2090, 2095, 2107, 2116, 2122, 2126, 2144, 2169, 2177, 2185, 2188, 2199, 2306, 2317, 2331, 2334, 2341, 2366, 2379, 2394, 2406, 2411, 2426, 2429, 2437, 2486, 2488, 2490, 2498, ...
61 | 2, 19, 69, 88, 138, 155, 205, 234, 306, 336, 420, 425, 455, 470, 525, 555, 561, 608, 626, 667, 674, 766, 779, 846, 851, 937, 971, 998, 1053, 1055, 1112, 1137, 1263, 1329, 1383, 1445, 1448, 1499, 1544, 1551, 1578, 1640, 1689, 1724, 1750, 1793, 1868, 1892, 2143, 2241, 2358, 2407, 2414, 2441, ...
62 | 2, 6, 10, 36, 65, 74, 78, 83, 106, 115, 120, 148, 154, 161, 163, 168, 176, 189, 194, 197, 266, 270, 288, 331, 385, 399, 407, 410, 412, 413, 431, 468, 513, 524, 546, 569, 572, 578, 581, 600, 611, 619, 625, 626, 647, 719, 723, 756, 832, 834, 849, 922, 986, 1006, 1007, 1047, 1064, 1102, 1199, 1217, 1247, 1332, 1335, 1349, 1353, 1372, 1374, 1382, 1400, 1405, 1410, 1418, 1419, 1422, 1426, 1444, 1456, 1457, 1554, 1568, 1590, 1622, 1678, 1680, 1683, 1684, 1695, 1725, 1797, 1840, 1847, 1861, 1874, 1895, 1939, 1950, 1987, 1988, 1990, 1999, 2070, 2081, 2093, 2180, 2183, 2212, 2216, 2220, 2227, 2266, 2303, 2337, 2349, 2389, 2408, 2418, 2426, ...
63 | 3, 7, 12, 40, 92, 122, 178, 191, 195, 214, 282, 283, 297, 299, 300, 418, 437, 467, 493, 497, 507, 512, 530, 597, 630, 664, 667, 701, 744, 800, 823, 876, 899, 944, 965, 1040, 1043, 1061, 1063, 1070, 1083, 1112, 1141, 1209, 1254, 1322, 1325, 1345, 1378, 1398, 1416, 1430, 1532, 1563, 1625, 1704, 1726, 1728, 1746, 1863, 1875, 1921, 1997, 2030, 2085, 2179, 2220, 2319, 2392, 2461, 2476, ...
64 | 3, 9, 21, 30, 54, 65, 75, 96, 112, 114, 132, 156, 163, 181, 191, 229, 251, 332, 342, 360, 363, 376, 428, 430, 432, 448, 527, 562, 583, 588, 589, 605, 726, 738, 763, 804, 831, 839, 847, 850, 884, 971, 1068, 1099, 1127, 1142, 1167, 1181, 1198, 1199, 1201, 1225, 1243, 1255, 1273, 1299, 1306, 1315, 1379, 1489, 1540, 1547, 1568, 1596, 1673, 1678, 1713, 1714, 1754, 1801, 1812, 1818, 1823, 1847, 1861, 1878, 1906, 1957, 1960, 1962, 1987, 2046, 2069, 2087, 2098, 2109, 2118, 2142, 2177, 2195, 2325, 2330, 2385, 2418, 2434, 2457, ...
65 | 2, 68, 71, 83, 87, 92, 132, 137, 316, 390, 394, 407, 451, 534, 553, 587, 597, 622, 652, 733, 735, 771, 781, 846, 871, 879, 892, 901, 943, 987, 1047, 1077, 1102, 1120, 1179, 1208, 1219, 1224, 1229, 1261, 1303, 1371, 1614, 1623, 1648, 1726, 1745, 1796, 1810, 1813, 1821, 1842, 1920, 1965, 2147, 2153, 2156, 2193, 2330, 2434, ...
66 | 9, 38, 40, 53, 62, 105, 107, 125, 129, 134, 156, 183, 193, 210, 276, 283, 294, 306, 307, 317, 345, 371, 388, 399, 447, 504, 508, 514, 529, 542, 585, 594, 648, 668, 671, 676, 717, 719, 722, 723, 743, 752, 772, 784, 818, 833, 840, 844, 870, 880, 896, 941, 942, 964, 978, 1021, 1087, 1115, 1144, 1168, 1182, 1194, 1212, 1215, 1239, 1366, 1378, 1527, 1532, 1562, 1603, 1630, 1644, 1712, 1718, 1780, 1831, 1851, 1864, 1881, 1882, 1885, 1890, 1929, 1941, 1943, 1949, 1957, 1980, 2003, 2036, 2052, 2110, 2124, 2158, 2226, 2269, 2317, 2331, 2344, 2366, 2388, 2422, 2452, ...
67 | 46, 122, 238, 304, 314, 315, 328, 332, 346, 372, 382, 426, 440, 491, 496, 510, 524, 528, 566, 638, 733, 826, 1016, 1054, 1071, 1214, 1309, 1338, 1388, 1401, 1457, 1512, 1536, 1582, 1624, 1718, 1773, 1814, 1816, 1825, 1952, 1985, 2021, 2072, 2308, 2349, 2449, 2481, ...
68 | 47, 85, 88, 94, 150, 175, 177, 190, 194, 197, 203, 210, 217, 233, 254, 259, 315, 316, 322, 345, 371, 405, 451, 500, 505, 510, 562, 572, 619, 658, 730, 759, 838, 889, 926, 959, 983, 1041, 1090, 1110, 1136, 1145, 1218, 1249, 1251, 1336, 1373, 1374, 1376, 1392, 1416, 1418, 1425, 1484, 1513, 1547, 1562, 1567, 1581, 1590, 1641, 1657, 1804, 1824, 1829, 1903, 1949, 1952, 1966, 1971, 1990, 2001, 2028, 2031, 2059, 2095, 2109, 2119, 2277, 2372, 2381, 2416, 2421, ...
69 | 2, 22, 26, 43, 82, 218, 271, 273, 386, 392, 412, 469, 589, 599, 616, 686, 689, 807, 842, 848, 855, 867, 1055, 1138, 1170, 1251, 1287, 1352, 1393, 1422, 1555, 1590, 1617, 1662, 1712, 1745, 1851, 1998, 1999, 2069, 2076, 2189, 2213, 2279, 2328, 2440, ...
70 | 3, 12, 16, 19, 24, 37, 50, 57, 110, 148, 166, 169, 174, 181, 192, 212, 228, 233, 242, 330, 370, 380, 469, 492, 505, 514, 576, 583, 655, 657, 676, 678, 716, 726, 734, 776, 786, 829, 857, 994, 1040, 1099, 1114, 1154, 1181, 1186, 1244, 1338, 1348, 1350, 1354, 1379, 1397, 1436, 1444, 1509, 1517, 1549, 1557, 1591, 1645, 1741, 1752, 1755, 1801, 1813, 1855, 1900, 1925, 1935, 1949, 1960, 1971, 2003, 2004, 2069, 2082, 2084, 2089, 2113, 2130, 2167, 2259, 2286, 2288, 2301, 2338, 2361, 2362, 2380, 2382, 2386, 2393, 2401, 2420, ...
71 | 3, 6, 17, 24, 37, 89, 132, 374, 387, 402, 421, 435, 453, 464, 490, 516, 708, 736, 919, 947, 981, 1033, 1067, 1170, 1195, 1253, 1284, 1349, 1385, 1409, 1479, 1709, 1724, 1726, 1735, 1875, 1950, 1984, 2012, 2070, 2124, 2133, 2161, 2194, 2202, 2424, 2432, 2459, ...
72 | 3, 18, 36, 43, 51, 77, 89, 150, 178, 195, 209, 219, 244, 254, 265, 280, 290, 294, 314, 322, 340, 341, 362, 363, 371, 377, 386, 416, 450, 474, 488, 494, 511, 546, 548, 554, 576, 611, 625, 647, 723, 740, 760, 766, 800, 801, 803, 807, 820, 827, 835, 883, 940, 943, 956, 976, 1013, 1041, 1105, 1161, 1193, 1198, 1217, 1257, 1321, 1333, 1379, 1380, 1405, 1430, 1472, 1550, 1555, 1568, 1698, 1729, 1736, 1750, 1828, 1829, 1841, 1844, 1950, 1952, 1954, 1994, 1998, 2060, 2074, 2123, 2129, 2160, 2174, 2225, 2233, 2244, 2284, 2286, 2392, 2401, 2407, 2411, 2417, 2428, 2473, 2488, ...
73 | 11, 15, 75, 114, 195, 215, 295, 335, 378, 559, 566, 650, 660, 832, 871, 904, 966, 1021, 1112, 1203, 1334, 1433, 1485, 1724, 1822, 1959, 1998, 2115, 2154, 2432, 2465, 2486, ...
74 | 16, 19, 21, 49, 56, 63, 71, 74, 77, 83, 92, 96, 99, 160, 172, 197, 198, 230, 241, 280, 283, 415, 425, 448, 490, 520, 627, 691, 735, 784, 803, 829, 842, 853, 871, 872, 893, 894, 973, 981, 989, 1043, 1060, 1061, 1071, 1179, 1182, 1203, 1290, 1299, 1317, 1370, 1389, 1477, 1478, 1496, 1562, 1601, 1624, 1714, 1789, 1848, 1890, 1942, 2016, 2083, 2156, 2217, 2250, 2253, 2266, 2313, 2315, 2319, 2323, 2343, 2358, 2361, 2363, 2383, 2396, 2398, 2399, 2401, 2486, ...
75 | 59, 73, 75, 165, 181, 204, 205, 208, 283, 369, 377, 396, 427, 448, 531, 535, 573, 587, 601, 628, 664, 686, 705, 727, 836, 860, 866, 879, 907, 918, 1077, 1118, 1142, 1175, 1220, 1256, 1284, 1332, 1337, 1349, 1389, 1412, 1426, 1437, 1453, 1493, 1499, 1502, 1551, 1561, 1640, 1737, 1748, 1789, 1796, 1958, 1975, 2056, 2089, 2090, 2143, 2158, 2177, 2210, 2212, 2245, 2300, 2306, 2347, 2371, 2415, 2424, 2491, ...
76 | 7, 29, 33, 39, 62, 69, 81, 82, 139, 165, 180, 185, 196, 253, 273, 274, 278, 337, 348, 358, 376, 389, 412, 419, 439, 507, 521, 533, 555, 571, 600, 605, 611, 621, 623, 626, 650, 658, 701, 714, 727, 733, 758, 767, 795, 817, 866, 868, 869, 878, 881, 885, 896, 905, 932, 949, 983, 1013, 1045, 1072, 1082, 1128, 1144, 1162, 1170, 1171, 1196, 1258, 1276, 1285, 1294, 1307, 1315, 1355, 1416, 1423, 1424, 1430, 1442, 1495, 1532, 1535, 1557, 1615, 1651, 1659, 1664, 1669, 1764, 1816, 1833, 1852, 1872, 1930, 1931, 1965, 2077, 2103, 2104, 2182, 2211, 2235, 2271, 2333, 2341, 2402, 2427, 2480, 2481, ...
77 | 2, 17, 41, 42, 90, 104, 130, 132, 186, 211, 224, 228, 235, 259, 307, 319, 362, 410, 412, 537, 603, 605, 783, 789, 795, 811, 814, 854, 858, 861, 877, 892, 906, 940, 1003, 1028, 1052, 1095, 1178, 1241, 1249, 1259, 1263, 1265, 1279, 1357, 1392, 1398, 1457, 1569, 1755, 1795, 1809, 1814, 1880, 1882, 1894, 1896, 1906, 1961, 1970, 2020, 2026, 2054, 2132, 2161, 2179, 2253, 2310, 2327, 2408, 2451, ...
78 | 2, 8, 13, 16, 18, 41, 87, 88, 123, 134, 137, 141, 166, 180, 198, 236, 270, 288, 293, 299, 321, 361, 405, 416, 422, 450, 457, 492, 495, 514, 558, 580, 602, 605, 673, 697, 712, 713, 724, 743, 784, 805, 879, 884, 893, 1013, 1017, 1049, 1067, 1094, 1117, 1157, 1212, 1231, 1272, 1306, 1331, 1437, 1447, 1462, 1493, 1528, 1542, 1570, 1572, 1589, 1624, 1676, 1726, 1809, 1818, 1819, 1830, 1843, 1900, 1904, 1929, 1965, 1998, 2000, 2027, 2031, 2065, 2133, 2152, 2159, 2206, 2252, 2374, 2414, 2425, 2448, 2449, 2451, 2462, 2471, 2499, ...
79 | 22, 112, 140, 158, 159, 170, 254, 271, 330, 334, 354, 390, 483, 528, 560, 565, 714, 850, 888, 924, 929, 933, 935, 970, 1019, 1047, 1141, 1265, 1266, 1338, 1359, 1376, 1412, 1485, 1504, 1542, 1598, 1607, 1618, 1747, 1773, 1814, 1843, 2055, 2087, 2088, 2100, 2167, 2186, 2233, 2311, 2358, 2359, 2363, 2429, ...
80 | 2, 12, 13, 53, 71, 97, 123, 149, 150, 172, 197, 202, 228, 252, 286, 310, 323, 331, 335, 338, 367, 378, 400, 401, 404, 488, 504, 550, 569, 589, 615, 657, 673, 681, 699, 722, 770, 779, 787, 795, 857, 995, 1002, 1004, 1006, 1008, 1054, 1063, 1115, 1129, 1138, 1183, 1187, 1196, 1200, 1201, 1227, 1235, 1266, 1272, 1295, 1314, 1342, 1383, 1402, 1407, 1416, 1421, 1435, 1439, 1461, 1476, 1480, 1510, 1514, 1525, 1554, 1568, 1596, 1620, 1626, 1640, 1658, 1686, 1695, 1719, 1763, 1894, 1942, 2008, 2020, 2051, 2060, 2132, 2139, 2144, 2183, 2199, 2220, 2244, 2246, 2263, 2397, 2468, 2471, ...
81 | 21, 28, 50, 77, 79, 142, 153, 191, 194, 310, 311, 405, 440, 462, 557, 562, 638, 659, 690, 724, 746, 852, 868, 887, 919, 922, 944, 945, 1140, 1183, 1326, 1344, 1452, 1463, 1486, 1507, 1579, 1607, 1632, 1652, 1659, 1683, 1710, 1788, 1801, 1812, 1851, 1891, 1925, 1943, 1992, 2157, 2294, 2309, 2352, 2402, 2473, ...
82 | 61, 63, 99, 144, 230, 312, 360, 401, 413, 424, 451, 515, 542, 567, 610, 618, 622, 651, 690, 732, 817, 871, 1007, 1100, 1106, 1156, 1278, 1403, 1427, 1460, 1535, 1557, 1572, 1604, 1681, 1742, 1802, 1820, 1847, 1903, 1910, 1913, 1978, 2019, 2104, 2134, 2152, 2169, 2309, 2383, 2491, ...
83 | 41, 146, 386, 593, 667, 688, 906, 927, 930, 1025, 1032, 1111, 1410, 1437, 1638, 1829, 1960, 2045, 2159, 2381, 2384, ...
84 | 7, 12, 53, 61, 70, 71, 91, 132, 142, 156, 165, 166, 173, 187, 209, 218, 219, 236, 244, 260, 266, 270, 276, 286, 287, 306, 315, 324, 351, 361, 400, 424, 452, 461, 501, 502, 549, 569, 613, 615, 630, 631, 632, 639, 659, 669, 698, 710, 718, 721, 728, 751, 757, 782, 794, 797, 798, 826, 828, 831, 851, 853, 878, 883, 893, 895, 915, 919, 947, 965, 981, 984, 986, 1055, 1078, 1119, 1160, 1175, 1183, 1186, 1188, 1206, 1260, 1267, 1280, 1308, 1310, 1326, 1329, 1341, 1342, 1364, 1401, 1419, 1436, 1438, 1473, 1488, 1506, 1510, 1539, 1550, 1593, 1612, 1630, 1653, 1659, 1663, 1684, 1687, 1705, 1708, 1746, 1754, 1765, 1766, 1768, 1770, 1777, 1839, 1884, 1911, 1918, 1967, 1984, 1987, 1990, 2011, 2045, 2050, 2061, 2084, 2103, 2123, 2156, 2177, 2181, 2229, 2240, 2245, 2246, 2250, 2256, 2259, 2265, 2269, 2291, 2307, 2309, 2310, 2349, 2358, 2376, 2391, 2392, 2427, 2481, 2491, 2492, ...
85 | 2, 15, 32, 65, 77, 116, 154, 173, 197, 233, 250, 267, 392, 430, 453, 516, 558, 740, 791, 826, 844, 860, 861, 867, 868, 963, 1009, 1010, 1048, 1055, 1097, 1198, 1200, 1220, 1243, 1290, 1329, 1434, 1619, 1621, 1623, 1652, 1710, 1743, 1745, 1827, 1836, 1880, 1932, 2059, 2091, 2109, 2196, ...
86 | 2, 3, 6, 22, 59, 83, 91, 95, 120, 148, 195, 196, 201, 247, 252, 257, 264, 315, 360, 378, 458, 555, 680, 792, 893, 1025, 1088, 1158, 1171, 1240, 1280, 1416, 1437, 1632, 1661, 1677, 1681, 1849, 1946, 1960, 2007, 2090, 2092, 2225, 2242, 2244, 2377, 2483, ...
87 | 8, 27, 33, 34, 59, 118, 119, 124, 155, 189, 249, 263, 270, 275, 286, 334, 342, 382, 402, 414, 421, 468, 474, 486, 501, 530, 563, 577, 655, 697, 764, 895, 915, 932, 1108, 1175, 1188, 1228, 1234, 1268, 1277, 1308, 1357, 1358, 1370, 1373, 1400, 1491, 1519, 1529, 1558, 1560, 1658, 1693, 1699, 1735, 1745, 1773, 1805, 1866, 1876, 1905, 1915, 1961, 1996, 2161, 2192, 2227, 2297, 2321, 2446, 2483, ...
88 | 5, 97, 131, 189, 199, 200, 258, 263, 289, 317, 352, 398, 470, 533, 554, 601, 603, 606, 732, 764, 765, 819, 869, 872, 888, 961, 1197, 1299, 1355, 1435, 1469, 1471, 1522, 1577, 1582, 1638, 1649, 1743, 1768, 1788, 1829, 1860, 1879, 1946, 1994, 2091, 2208, 2289, 2297, 2302, 2322, 2378, 2401, 2475, 2487, ...
89 | 2, 114, 159, 190, 234, 251, 436, 616, 834, 878, 1008, 1049, 1060, 1062, 1118, 1472, 1689, 1792, 2282, 2334, 2463, 2494, ...
90 | 2, 5, 6, 58, 66, 71, 78, 80, 96, 101, 122, 127, 148, 153, 183, 194, 198, 211, 244, 258, 261, 354, 387, 390, 413, 415, 446, 447, 450, 465, 475, 480, 502, 524, 532, 549, 577, 590, 607, 608, 610, 620, 623, 635, 668, 678, 716, 793, 844, 859, 862, 879, 895, 915, 958, 963, 969, 974, 1009, 1035, 1080, 1093, 1095, 1136, 1137, 1178, 1219, 1225, 1230, 1296, 1301, 1309, 1315, 1352, 1359, 1369, 1375, 1381, 1383, 1389, 1411, 1450, 1453, 1463, 1514, 1599, 1638, 1651, 1682, 1692, 1705, 1710, 1734, 1768, 1803, 1850, 1883, 1936, 1965, 2037, 2041, 2066, 2108, 2111, 2114, 2117, 2120, 2145, 2164, 2195, 2210, 2225, 2287, 2293, 2327, 2382, 2387, 2389, 2406, 2407, 2458, 2470, 2472, 2483, 2485, 2498, ...
91 | 11, 19, 28, 30, 46, 80, 116, 140, 159, 198, 272, 331, 357, 360, 448, 544, 728, 827, 854, 910, 937, 1019, 1041, 1106, 1164, 1174, 1226, 1229, 1287, 1307, 1327, 1348, 1408, 1471, 1473, 1618, 1642, 1751, 1810, 1900, 1965, 1976, 2051, 2110, 2206, 2226, 2254, 2331, ...
92 | 4, 5, 12, 19, 43, 47, 70, 164, 183, 225, 258, 261, 287, 297, 306, 323, 335, 340, 351, 363, 383, 397, 442, 534, 545, 558, 572, 594, 624, 652, 656, 659, 666, 673, 681, 696, 701, 724, 730, 773, 810, 887, 923, 933, 972, 997, 1002, 1006, 1061, 1124, 1135, 1178, 1196, 1225, 1230, 1313, 1356, 1407, 1409, 1451, 1452, 1483, 1576, 1586, 1602, 1607, 1612, 1619, 1667, 1691, 1697, 1823, 1827, 1935, 1963, 1992, 2035, 2099, 2138, 2156, 2199, 2252, 2274, 2299, 2393, 2398, 2447, 2477, ...
93 | 2, 10, 15, 22, 27, 63, 98, 113, 142, 156, 188, 236, 278, 282, 293, 300, 315, 333, 353, 374, 420, 487, 509, 558, 580, 592, 793, 924, 1099, 1131, 1195, 1242, 1268, 1274, 1321, 1357, 1413, 1484, 1542, 1547, 1611, 1683, 1699, 1702, 1711, 1730, 1746, 1760, 1819, 1854, 1895, 1902, 2026, 2077, 2092, 2111, 2151, 2222, 2272, 2328, 2388, 2390, 2409, 2450, 2475, 2482, 2483, ...
94 | 6, 7, 17, 90, 126, 139, 143, 257, 293, 295, 319, 328, 387, 482, 519, 603, 720, 751, 819, 884, 896, 903, 905, 921, 952, 954, 956, 1042, 1058, 1147, 1170, 1237, 1253, 1279, 1295, 1343, 1366, 1370, 1406, 1465, 1514, 1593, 1595, 1607, 1609, 1622, 1701, 1705, 1709, 1736, 1772, 1791, 1833, 1860, 1866, 1879, 1969, 1990, 1992, 2059, 2115, 2132, 2169, 2219, 2233, 2253, 2344, ...
95 | 44, 47, 66, 155, 187, 344, 345, 448, 492, 525, 546, 611, 642, 704, 721, 748, 820, 858, 1062, 1108, 1112, 1195, 1277, 1314, 1453, 1558, 1597, 1667, 1725, 1760, 1780, 1793, 1807, 1901, 1963, 1983, 2021, 2055, 2060, 2251, 2266, 2331, ...
96 | 4, 19, 28, 36, 44, 79, 94, 106, 109, 125, 132, 147, 319, 343, 399, 412, 415, 435, 441, 449, 463, 491, 562, 576, 625, 643, 646, 649, 691, 701, 712, 787, 788, 820, 824, 840, 851, 976, 994, 1047, 1079, 1111, 1112, 1136, 1195, 1216, 1219, 1225, 1291, 1309, 1339, 1359, 1425, 1456, 1466, 1520, 1525, 1533, 1540, 1543, 1601, 1633, 1668, 1679, 1738, 1744, 1746, 1789, 1854, 1873, 1922, 2056, 2101, 2110, 2159, 2225, 2266, 2267, 2286, 2301, 2320, 2346, 2378, 2395, 2424, 2468, 2488, ...
97 | 12, 90, 104, 234, 271, 339, 420, 421, 428, 429, 464, 805, 909, 934, 1054, 1114, 1116, 1128, 1144, 1159, 1193, 1364, 1788, 2086, 2215, 2254, 2448, 2461, ...
98 | 2, 19, 20, 40, 52, 75, 80, 88, 95, 98, 100, 103, 106, 123, 124, 162, 176, 198, 220, 276, 311, 358, 396, 427, 429, 522, 525, 529, 555, 576, 586, 605, 675, 695, 733, 782, 811, 833, 872, 879, 891, 898, 950, 952, 971, 977, 982, 1076, 1090, 1150, 1156, 1219, 1231, 1257, 1260, 1312, 1340, 1483, 1621, 1707, 1716, 1742, 1774, 1806, 1810, 1847, 1850, 1926, 1935, 1975, 2019, 2042, 2090, 2153, 2167, 2169, 2238, 2260, 2262, 2306, 2346, 2356, 2451, 2474, ...
99 | 63, 88, 113, 123, 134, 146, 217, 284, 320, 352, 504, 511, 532, 544, 603, 704, 802, 840, 870, 974, 1205, 1242, 1278, 1414, 1441, 1471, 1516, 1579, 1660, 1740, 1763, 1765, 1846, 1874, 1889, 1932, 1939, 1980, 2095, 2132, 2136, 2396, 2493, ...
100 | 20, 22, 88, 106, 114, 133, 179, 189, 222, 224, 234, 312, 326, 333, 355, 440, 443, 492, 509, 535, 615, 676, 727, 751, 765, 767, 812, 813, 845, 925, 988, 1032, 1089, 1106, 1229, 1249, 1259, 1281, 1323, 1343, 1434, 1435, 1461, 1482, 1536, 1621, 1649, 1678, 1692, 1739, 1779, 1824, 1979, 2020, 2021, 2033, 2052, 2164, 2186, 2322, 2391, 2418, 2438, 2457, ...
prime p | bases such that p is unique prime

|

2 | 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, ...
3 | 2, 4, 5, 8, 10, 11, 17, 23, 26, 28, 35, 47, 53, 71, 80, 82, 95, 107, 143, 161, 191, 215, 242, 244, 287, 323, 383, 431, 485, 575, 647, 728, 730, 767, 863, 971, 1151, 1295, 1457, 1535, 1727, 1943, 2186, 2188, 2303, 2591, 2915, 3071, 3455, 3887, 4373, 4607, 5183, 5831, 6143, 6560, 6562, 6911, 7775, 8747, 9215, 10367, 11663, 12287, 13121, 13823, 15551, 17495, 18431, 19682, 19684, 20735, 23327, 24575, 26243, 27647, 31103, 34991, 36863, 39365, 41471, 46655, 49151, 52487, 55295, 59048, 59050, 62207, 69983, 73727, 78731, 82943, 93311, 98303, 104975, 110591, 118097, 124415, 139967, 147455, 157463, 165887, 177146, 177148, 186623, 196607, 209951, 221183, 236195, 248831, 279935, 294911, 314927, 331775, 354293, 373247, 393215, 419903, 442367, 472391, 497663, 531440, 531442, 559871, 589823, 629855, 663551, 708587, 746495, 786431, 839807, 884735, 944783, 995327, ...
5 | 2, 3, 4, 6, 7, 9, 19, 24, 26, 39, 49, 79, 99, 124, 126, 159, 199, 249, 319, 399, 499, 624, 626, 639, 799, 999, 1249, 1279, 1599, 1999, 2499, 2559, 3124, 3126, 3199, 3999, 4999, 5119, 6249, 6399, 7999, 9999, 10239, 12499, 12799, 15624, 15626, 15999, 19999, 20479, 24999, 25599, 31249, 31999, 39999, 40959, 49999, 51199, 62499, 63999, 78124, 78126, 79999, 81919, 99999, 102399, 124999, 127999, 156249, 159999, 163839, 199999, 204799, 249999, 255999, 312499, 319999, 327679, 390624, 390626, 399999, 409599, 499999, 511999, 624999, 639999, 655359, 781249, 799999, 819199, 999999, ...
7 | 2, 3, 4, 5, 6, 8, 13, 18, 19, 27, 48, 50, 55, 97, 111, 195, 223, 342, 344, 391, 447, 685, 783, 895, 1371, 1567, 1791, 2400, 2402, 2743, 3135, 3583, 4801, 5487, 6271, 7167, 9603, 10975, 12543, 14335, 16806, 16808, 19207, 21951, 25087, 28671, 33613, 38415, 43903, 50175, 57343, 67227, 76831, 87807, 100351, 114687, 117648, 117650, 134455, 153663, 175615, 200703, 229375, 235297, 268911, 307327, 351231, 401407, 458751, 470595, 537823, 614655, 702463, 802815, 823542, 823544, 917503, 941191, ...
11 | 2, 3, 10, 12, 21, 43, 87, 120, 122, 175, 241, 351, 483, 703, 967, 1330, 1332, 1407, 1935, 2661, 2815, 3871, 5323, 5631, 7743, 10647, 11263, 14640, 14642, 15487, 21295, 22527, 29281, 30975, 42591, 45055, 58563, 61951, 85183, 90111, 117127, 123903, 161050, 161052, 170367, 180223, 234255, 247807, 322101, 340735, 360447, 468511, 495615, 644203, 681471, 720895, 937023, 991231, ...
13 | 2, 3, 4, 5, 12, 14, 22, 23, 25, 51, 103, 168, 170, 207, 239, 337, 415, 675, 831, 1351, 1663, 2196, 2198, 2703, 3327, 4393, 5407, 6655, 8787, 10815, 13311, 17575, 21631, 26623, 28560, 28562, 35151, 43263, 53247, 57121, 70303, 86527, 106495, 114243, 140607, 173055, 212991, 228487, 281215, 346111, 371292, 371294, 425983, 456975, 562431, 692223, 742585, 851967, 913951, ...
17 | 2, 4, 16, 18, 33, 67, 135, 271, 288, 290, 543, 577, 1087, 1155, 2175, 2311, 4351, 4623, 4912, 4914, 8703, 9247, 9825, 17407, 18495, 19651, 34815, 36991, 39303, 69631, 73983, 78607, 83520, 83522, 139263, 147967, 157215, 167041, 278527, 295935, 314431, 334083, 557055, 591871, 628863, 668167, ...
19 | 2, 7, 8, 18, 20, 37, 75, 151, 303, 360, 362, 607, 721, 1215, 1443, 2431, 2887, 4863, 5775, 6858, 6860, 9727, 11551, 13717, 19455, 23103, 27435, 38911, 46207, 54871, 77823, 92415, 109743, 130320, 130322, 155647, 184831, 219487, 260641, 311295, 369663, 438975, 521283, 622591, 739327, 877951, ...
23 | 22, 24, 45, 91, 183, 367, 528, 530, 735, 1057, 1471, 2115, 2943, 4231, 5887, 8463, 11775, 12166, 12168, 16927, 23551, 24333, 33855, 47103, 48667, 67711, 94207, 97335, 135423, 188415, 194671, 270847, 279840, 279842, 376831, 389343, 541695, 559681, 753663, 778687, ...
29 | 28, 30, 41, 57, 115, 231, 463, 840, 842, 927, 1681, 1855, 3363, 3711, 6727, 7423, 13455, 14847, 24388, 24390, 26911, 29695, 48777, 53823, 59391, 97555, 107647, 118783, 195111, 215295, 237567, 390223, 430591, 475135, 707280, 707282, 780447, 861183, 950271, ...
31 | 2, 5, 6, 30, 32, 61, 123, 247, 495, 960, 962, 991, 1921, 1983, 3843, 3967, 7687, 7935, 15375, 15871, 29790, 29792, 30751, 31743, 59581, 61503, 63487, 119163, 123007, 126975, 238327, 246015, 253951, 476655, 492031, 507903, 923520, 923522, 953311, 984063, ...
37 | 6, 10, 11, 36, 38, 73, 147, 295, 591, 1183, 1368, 1370, 2367, 2737, 4735, 5475, 9471, 10951, 18943, 21903, 37887, 43807, 50652, 50654, 75775, 87615, 101305, 151551, 175231, 202611, 303103, 350463, 405223, 606207, 700927, 810447, ...
41 | 2, 3, 4, 9, 40, 42, 81, 163, 327, 655, 1311, 1680, 1682, 2623, 3361, 5247, 6723, 10495, 13447, 20991, 26895, 41983, 53791, 68920, 68922, 83967, 107583, 137841, 167935, 215167, 275683, 335871, 430335, 551367, 671743, 860671, ...
43 | 2, 6, 7, 42, 44, 85, 171, 343, 687, 1375, 1848, 1850, 2751, 3697, 5503, 7395, 11007, 14791, 22015, 29583, 44031, 59167, 79506, 79508, 88063, 118335, 159013, 176127, 236671, 318027, 352255, 473343, 636055, 704511, 946687, ...
47 | 46, 48, 93, 187, 375, 751, 1503, 2208, 2210, 3007, 4417, 6015, 8835, 12031, 17671, 24063, 35343, 48127, 70687, 96255, 103822, 103824, 141375, 192511, 207645, 282751, 385023, 415291, 565503, 770047, 830583, ...
53 | 52, 54, 105, 211, 423, 847, 1695, 2808, 2810, 3391, 5617, 6783, 11235, 13567, 22471, 27135, 44943, 54271, 89887, 108543, 148876, 148878, 179775, 217087, 297753, 359551, 434175, 595507, 719103, 868351, ...
59 | 58, 60, 117, 235, 471, 943, 1887, 3480, 3482, 3775, 6961, 7551, 13923, 15103, 27847, 30207, 55695, 60415, 111391, 120831, 205378, 205380, 222783, 241663, 410757, 445567, 483327, 821515, 891135, 966655, ...
61 | 3, 11, 13, 14, 60, 62, 121, 243, 487, 975, 1951, 3720, 3722, 3903, 7441, 7807, 14883, 15615, 29767, 31231, 59535, 62463, 119071, 124927, 226980, 226982, 238143, 249855, 453961, 476287, 499711, 907923, 952575, 999423, ...
67 | 66, 68, 133, 267, 535, 1071, 2143, 4287, 4488, 4490, 8575, 8977, 17151, 17955, 34303, 35911, 68607, 71823, 137215, 143647, 274431, 287295, 300762, 300764, 548863, 574591, 601525, ...
71 | 70, 72, 141, 283, 567, 1135, 2271, 4543, 5040, 5042, 9087, 10081, 18175, 20163, 36351, 40327, 72703, 80655, 145407, 161311, 290815, 322623, 357910, 357912, 581631, 645247, 715821, ...
73 | 2, 3, 8, 9, 72, 74, 145, 291, 583, 1167, 2335, 4671, 5328, 5330, 9343, 10657, 18687, 21315, 37375, 42631, 74751, 85263, 149503, 170527, 299007, 341055, 389016, 389018, 598015, 682111, 778033, ...
79 | 78, 80, 157, 315, 631, 1263, 2527, 5055, 6240, 6242, 10111, 12481, 20223, 24963, 40447, 49927, 80895, 99855, 161791, 199711, 323583, 399423, 493038, 493040, 647167, 798847, 986077, ...
83 | 82, 84, 165, 331, 663, 1327, 2655, 5311, 6888, 6890, 10623, 13777, 21247, 27555, 42495, 55111, 84991, 110223, 169983, 220447, 339967, 440895, 571786, 571788, 679935, 881791, ...
89 | 88, 90, 177, 355, 711, 1423, 2847, 5695, 7920, 7922, 11391, 15841, 22783, 31683, 45567, 63367, 91135, 126735, 182271, 253471, 364543, 506943, 704968, 704970, 729087, ...
97 | 96, 98, 193, 387, 775, 1551, 3103, 6207, 9408, 9410, 12415, 18817, 24831, 37635, 49663, 75271, 99327, 150543, 198655, 301087, 397311, 602175, 794623, 912672, 912674, ...

## Bibliography

* Chris K. Caldwell, Harvey Dubner, "Unique-period primes", Journal of Recreational Mathematics 29:1:43-48 (1998) preprint

## References

1. ↑ Caldwell, Chris. "Unique prime". The Prime Pages. http://primes.utm.edu/glossary/xpage/UniquePrime.html.
2. ↑ PRP Records: Probable Primes Top 10000
3. ↑ The Top Twenty Unique; Chris Caldwell
4. ↑ PRP records
5. ↑ The Cunningham Project
6. ↑ PRP records

* Yates, Samuel (1980). "Periods of unique primes". Math. Mag. 53: 314.

## External links

* Prime Top Tens
* Unique Period Primes
* Factorization of repunit numbers

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Prime number classes

By formula|

* Fermat (22n \+ 1)
* Mersenne (2p − 1)
* Double Mersenne (22p−1 − 1)
* Wagstaff (2p \+ 1)/3
* Proth (k·2n \+ 1)
* Factorial (n! ± 1)
* Primorial (pn# ± 1)
* Euclid (pn# + 1)
* Pythagorean (4n \+ 1)
* Pierpont (2m·3n \+ 1)
* Quartan (x4 \+ y4)
* Solinas (2m ± 2n ± 1)
* Cullen (n·2n \+ 1)
* Woodall (n·2n − 1)
* Cuban (x3 − y3)/(x − y)
* Carol (2n − 1)2 − 2
* Kynea (2n \+ 1)2 − 2
* Leyland (xy \+ yx)
* Thabit (3·2n − 1)
* Williams ((b−1)·bn − 1)
* Mills (⌊A3n⌋)

By integer sequence|

* Fibonacci
* Lucas
* Pell
* Newman–Shanks–Williams
* Perrin
* Partitions
* Bell
* Motzkin

By property|

* Wieferich (pair)
* Wall–Sun–Sun
* Wolstenholme
* Wilson
* Lucky
* Fortunate
* Ramanujan
* Pillai
* Regular
* Strong
* Stern
* Supersingular (elliptic curve)
* Supersingular (moonshine theory)
* Good
* Super
* Higgs
* Highly cototient

Base-dependent|

* Happy
* Dihedral
* Palindromic
* Emirp
* Repunit (10n − 1)/9
* Permutable
* Circular
* Truncatable
* Strobogrammatic
* Minimal
* Weakly
* Full reptend
* Unique
* Primeval
* Self
* Smarandache–Wellin
* Tetradic

Patterns|

* Twin (p, p \+ 2)
* Bi-twin chain (n − 1, n \+ 1, 2n − 1, 2n \+ 1, …)
* Triplet (p, p \+ 2 or p \+ 4, p \+ 6)
* Quadruplet (p, p \+ 2, p \+ 6, p \+ 8)
* k−Tuple
* Cousin (p, p \+ 4)
* Sexy (p, p \+ 6)
* Chen
* Sophie Germain (p, 2p + 1)
* Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...)
* Safe (p, (p − 1)/2)
* Arithmetic progression (p \+ a·n, n = 0, 1, 2, 3, ...)
* Balanced (consecutive p − n, p, p \+ n)

By size|

* Titanic (1,000+ digits)
* Gigantic (10,000+ digits)
* Mega (1,000,000+ digits)
* Largest known

Complex numbers|

* Eisenstein prime
* Gaussian prime

Composite numbers|

* Pseudoprime
* Catalan
* Elliptic
* Euler
* Euler–Jacobi
* Fermat
* Frobenius
* Lucas
* Somer–Lucas
* Strong
* Carmichael number
* Almost prime
* Semiprime
* Interprime
* Pernicious

Related topics|

* Probable prime
* Industrial-grade prime
* Illegal prime
* Formula for primes
* Prime gap

First 60 primes|

* 2
* 3
* 5
* 7
* 11
* 13
* 17
* 19
* 23
* 29
* 31
* 37
* 41
* 43
* 47
* 53
* 59
* 61
* 67
* 71
* 73
* 79
* 83
* 89
* 97
* 101
* 103
* 107
* 109
* 113
* 127
* 131
* 137
* 139
* 149
* 151
* 157
* 163
* 167
* 173
* 179
* 181
* 191
* 193
* 197
* 199
* 211
* 223
* 227
* 229
* 233
* 239
* 241
* 251
* 257
* 263
* 269
* 271
* 277
* 281

List of prime numbers

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