List of integer sequences

Short description: None

This is a list of notable integer sequences with links to their entries in the On-Line Encyclopedia of Integer Sequences.

## Contents

* 1 General
* 2 Figurate numbers
* 3 Types of primes
* 4 Base-dependent
* 5 References
* 6 External links

## General

Name | First elements | Short description | OEIS

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Kolakoski sequence | 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, ...  | The nth term describes the length of the nth run  | A000002
Euler's totient function φ(n) | 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, ...  | φ(n) is the number of positive integers not greater than n that are coprime with n.  | A000010
Lucas numbers L(n) | 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, ...  | L(n) = L(n − 1) + L(n − 2) for n ≥ 2, with L(0) = 2 and L(1) = 1.  | A000032
Prime numbers pn | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...  | The prime numbers pn, with n ≥ 1. A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.  | A000040
Partition numbers
Pn | 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...  | The partition numbers, number of additive breakdowns of n.  | A000041
Fibonacci numbers F(n) | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...  | F(n) = F(n − 1) + F(n − 2) for n ≥ 2, with F(0) = 0 and F(1) = 1.  | A000045
Sylvester's sequence | 2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, ...  | a(n \+ 1) = a(n)⋅a(n − 1)⋅ ⋯ ⋅a(0) + 1 = a(n)2 − a(n) + 1 for n ≥ 1, with a(0) = 2.  | A000058
Tribonacci numbers  | 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, ...  | T(n) = T(n − 1) + T(n − 2) + T(n − 3) for n ≥ 3, with T(0) = 0 and T(1) = T(2) = 1.  | A000073
Powers of 2 | 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...  | Powers of 2: 2n for n ≥ 0  | A000079
Polyominoes  | 1, 1, 1, 2, 5, 12, 35, 108, 369, ...  | The number of free polyominoes with n cells.  | A000105
Catalan numbers Cn | 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, ...  | Cn=1n+1(2nn)=(2n)!(n+1)!n!=∏k=2nn+kk,n≥0. | A000108
Bell numbers Bn | 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, ...  | Bn is the number of partitions of a set with n elements.  | A000110
Euler zigzag numbers En | 1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, ...  | En is the number of linear extensions of the "zig-zag" poset.  | A000111
Lazy caterer's sequence | 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, ...  | The maximal number of pieces formed when slicing a pancake with n cuts.  | A000124
Pell numbers Pn | 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, ...  | a(n) = 2a(n − 1) + a(n − 2) for n ≥ 2, with a(0) = 0, a(1) = 1.  | A000129
Factorials n! | 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, ...  | n! = 1⋅2⋅3⋅4⋅ ⋯ ⋅n for n ≥ 1, with 0! = 1 (empty product).  | A000142
Derangements | 1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, ...  | Number of permutations of n elements with no fixed points.  | A000166
Divisor function σ(n) | 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, ...  | σ(n) := σ1(n) is the sum of divisors of a positive integer n.  | A000203
Fermat numbers Fn | 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, ...  | Fn = 22n \+ 1 for n ≥ 0.  | A000215
Polytrees | 1, 1, 3, 8, 27, 91, 350, 1376, 5743, 24635, 108968, ...  | Number of oriented trees with n nodes.  | A000238
Perfect numbers | 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, ...  | n is equal to the sum s(n) = σ(n) − n of the proper divisors of n.  | A000396
Ramanujan tau function | 1, −24, 252, −1472, 4830, −6048, −16744, 84480, −113643, ...  | Values of the Ramanujan tau function, τ(n) at n = 1, 2, 3, ...  | A000594
Landau's function | 1, 1, 2, 3, 4, 6, 6, 12, 15, 20, ...  | The largest order of permutation of n elements.  | A000793
Narayana's cows  | 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, ...  | The number of cows each year if each cow has one cow a year beginning its fourth year.  | A000930
Padovan sequence | 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, ...  | P(n) = P(n − 2) + P(n − 3) for n ≥ 3, with P(0) = P(1) = P(2) = 1.  | A000931
Euclid–Mullin sequence | 2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, ...  | a(1) = 2; a(n \+ 1) is smallest prime factor of a(1) a(2) ⋯ a(n) + 1.  | A000945
Lucky numbers | 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, ...  | A natural number in a set that is filtered by a sieve.  | A000959
Prime powers | 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, ...  | Positive integer powers of prime numbers  | A000961
Central binomial coefficients | 1, 2, 6, 20, 70, 252, 924, ...  | (2nn)=(2n)!(n!)2 for all n≥0, numbers in the center of even rows of Pascal's triangle | A000984
Motzkin numbers | 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, ...  | The number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle.  | A001006
Jordan–Pólya numbers | 1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, ...  | Numbers that are the product of factorials.  | A001013
Jacobsthal numbers | 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, ...  | a(n) = a(n − 1) + 2a(n − 2) for n ≥ 2, with a(0) = 0, a(1) = 1.  | A001045
Sum of proper divisors s(n) | 0, 1, 1, 3, 1, 6, 1, 7, 4, 8, ...  | s(n) = σ(n) − n is the sum of the proper divisors of the positive integer n.  | A001065
Wedderburn–Etherington numbers | 0, 1, 1, 1, 2, 3, 6, 11, 23, 46, ...  | The number of binary rooted trees (every node has out-degree 0 or 2) with n endpoints (and 2n − 1 nodes in all).  | A001190
Gould's sequence | 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, ...  | Number of odd entries in row n of Pascal's triangle.  | A001316
Semiprimes | 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ...  | Products of two primes, not necessarily distinct.  | A001358
Golomb sequence | 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, ...  | a(n) is the number of times n occurs, starting with a(1) = 1.  | A001462
Perrin numbers Pn | 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, ...  | P(n) = P(n − 2) + P(n − 3) for n ≥ 3, with P(0) = 3, P(1) = 0, P(2) = 2.  | A001608
Sorting number | 0, 1, 3, 5, 8, 11, 14, 17, 21, 25, 29, 33, 37, 41, 45, 49, ...  | Used in the analysis of comparison sorts.  | A001855
Cullen numbers Cn | 1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, ...  | Cn = n⋅2n \+ 1, with n ≥ 0.  | A002064
Primorials pn# | 1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, ...  | pn#, the product of the first n primes.  | A002110
Highly composite numbers | 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ...  | A positive integer with more divisors than any smaller positive integer.  | A002182
Superior highly composite numbers | 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, ...  | A positive integer n for which there is an e > 0 such that  ⁠d(n)/ne⁠ ≥ ⁠d(k)/ke⁠ for all k > 1.  | A002201
Pronic numbers | 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, ...  | a(n) = 2t(n) = n(n \+ 1), with n ≥ 0 where t(n) are the triangular numbers.  | A002378
Markov numbers | 1, 2, 5, 13, 29, 34, 89, 169, 194, ...  | Positive integer solutions of x2 \+ y2 \+ z2 = 3xyz.  | A002559
Composite numbers | 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ...  | The numbers n of the form xy for x > 1 and y > 1.  | A002808
Ulam number | 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, ...  | a(1) = 1; a(2) = 2; for n > 2, a(n) is least number  > a(n − 1) which is a unique sum of two distinct earlier terms; semiperfect.  | A002858
Prime knots | 0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, ...  | The number of prime knots with n crossings.  | A002863
Carmichael numbers | 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, ...  | Composite numbers n such that an − 1 ≡ 1 (mod n) if a is coprime with n.  | A002997
Woodall numbers | 1, 7, 23, 63, 159, 383, 895, 2047, 4607, ...  | n⋅2n − 1, with n ≥ 1.  | A003261
Arithmetic numbers | 1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 27, ...  | An integer for which the average of its positive divisors is also an integer.  | A003601
Colossally abundant numbers | 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, ...  | A number n is colossally abundant if there is an ε > 0 such that for all k > 1,

σ(n)n1+ε≥σ(k)k1+ε,

where σ denotes the sum-of-divisors function.

| A004490
Alcuin's sequence | 0, 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, ...  | Number of triangles with integer sides and perimeter n.  | A005044
Deficient numbers | 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, ...  | Positive integers n such that σ(n) < 2n.  | A005100
Abundant numbers | 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, ...  | Positive integers n such that σ(n) > 2n.  | A005101
Untouchable numbers | 2, 5, 52, 88, 96, 120, 124, 146, 162, 188, ...  | Cannot be expressed as the sum of all the proper divisors of any positive integer.  | A005114
Recamán's sequence | 0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, ...  | "subtract if possible, otherwise add": a(0) = 0; for n > 0, a(n) = a(n − 1) − n if that number is positive and not already in the sequence, otherwise a(n) = a(n − 1) + n, whether or not that number is already in the sequence.  | A005132
Look-and-say sequence | 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, 13211311123113112211, ...  | A = 'frequency' followed by 'digit'-indication.  | A005150
Practical numbers | 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, ...  | All smaller positive integers can be represented as sums of distinct factors of the number.  | A005153
Alternating factorial | 1, 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019, ...  | n! − (n−1)! + (n−2)! − ... ± 1!.  | A005165
Fortunate numbers | 3, 5, 7, 13, 23, 17, 19, 23, 37, 61, ...  | The smallest integer m > 1 such that pn# + m is a prime number, where the primorial pn# is the product of the first n prime numbers.  | A005235
Semiperfect numbers | 6, 12, 18, 20, 24, 28, 30, 36, 40, 42, ...  | A natural number n that is equal to the sum of all or some of its proper divisors.  | A005835
Magic constants | 15, 34, 65, 111, 175, 260, 369, 505, 671, 870, 1105, 1379, 1695, 2056, ...  | Sum of numbers in any row, column, or diagonal of a magic square of order n ≥ 3.  | A006003
Weird numbers | 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, ...  | A natural number that is abundant but not semiperfect.  | A006037
Farey sequence numerators  | 0, 1, 0, 1, 1, 0, 1, 1, 2, 1, ...  |  | A006842
Farey sequence denominators  | 1, 1, 1, 2, 1, 1, 3, 2, 3, 1, ...  |  | A006843
Euclid numbers | 2, 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, ...  | pn# + 1, i.e. 1 + product of first n consecutive primes.  | A006862
Kaprekar numbers | 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, ...  | X2 = Abn \+ B, where 0 < B < bn and X = A \+ B.  | A006886
Sphenic numbers | 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, ...  | Products of 3 distinct primes.  | A007304
Giuga numbers | 30, 858, 1722, 66198, 2214408306, ...  | Composite numbers so that for each of its distinct prime factors pi we have pi2|(n−pi).  | A007850
Radical of an integer  | 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, ...  | The radical of a positive integer n is the product of the distinct prime numbers dividing n.  | A007947
Thue–Morse sequence | 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, ...  |  | A010060
Regular paperfolding sequence | 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, ...  | At each stage an alternating sequence of 1s and 0s is inserted between the terms of the previous sequence.  | A014577
Blum integers | 21, 33, 57, 69, 77, 93, 129, 133, 141, 161, 177, ...  | Numbers of the form pq where p and q are distinct primes congruent to 3 (mod 4).  | A016105
Magic numbers | 2, 8, 20, 28, 50, 82, 126, ...  | A number of nucleons (either protons or neutrons) such that they are arranged into complete shells within the atomic nucleus.  | A018226
Superperfect numbers | 2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056, ...  | Positive integers n for which σ2(n) = σ(σ(n)) = 2n. | A019279
Bernoulli numbers Bn | 1, −1, 1, 0, −1, 0, 1, 0, −1, 0, 5, 0, −691, 0, 7, 0, −3617, 0, 43867, 0, ...  |  | A027641
Hyperperfect numbers | 6, 21, 28, 301, 325, 496, 697, ...  | k-hyperperfect numbers, i.e. n for which the equality n = 1 + k (σ(n) − n − 1) holds.  | A034897
Achilles numbers | 72, 108, 200, 288, 392, 432, 500, 648, 675, 800, ...  | Positive integers which are powerful but imperfect.  | A052486
Primary pseudoperfect numbers | 2, 6, 42, 1806, 47058, 2214502422, 52495396602, ...  | Satisfies a certain Egyptian fraction.  | A054377
Erdős–Woods numbers | 16, 22, 34, 36, 46, 56, 64, 66, 70, 76, 78, 86, 88, ...  | The length of an interval of consecutive integers with property that every element has a factor in common with one of the endpoints.  | A059756
Sierpinski numbers | 78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, ...  | Odd k for which  { k⋅2n \+ 1 : n ∈ ℕ }  consists only of composite numbers.  | A076336
Riesel numbers | 509203, 762701, 777149, 790841, 992077, ...  | Odd k for which  { k⋅2n − 1 : n ∈ ℕ }  consists only of composite numbers.  | A076337
Baum–Sweet sequence | 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, ...  | a(n) = 1 if the binary representation of n contains no block of consecutive zeros of odd length; otherwise a(n) = 0.  | A086747
Gijswijt's sequence | 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, ...  | The nth term counts the maximal number of repeated blocks at the end of the subsequence from 1 to n−1 | A090822
Carol numbers  | −1, 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527, ...  | a(n)=(2n−1)2−2. | A093112
Juggler sequence | 0, 1, 1, 5, 2, 11, 2, 18, 2, 27, ...  | If n ≡ 0 (mod 2) then ⌊√n⌋ else ⌊n3/2⌋.  | A094683
Highly totient numbers | 1, 2, 4, 8, 12, 24, 48, 72, 144, 240, ...  | Each number k on this list has more solutions to the equation φ(x) = k than any preceding k.  | A097942
Euler numbers | 1, 0, −1, 0, 5, 0, −61, 0, 1385, 0, ...  | 1cosh⁡t=2et+e−t=∑n=0∞Enn!⋅tn. | A122045
Polite numbers | 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, ...  | A positive integer that can be written as the sum of two or more consecutive positive integers.  | A138591
Erdős–Nicolas numbers | 24, 2016, 8190, 42336, 45864, 392448, 714240, 1571328, ...  | A number n such that there exists another number m and ∑d∣n, d≤md=n. | A194472
Solution to Stepping Stone Puzzle  | 1, 16, 28, 38, 49, 60, ...  | The maximal value a(n) of the stepping stone puzzle  | A337663

## Figurate numbers

Main article: Figurate number

Name | First elements | Short description | OEIS

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Natural numbers | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...  | The natural numbers (positive integers) n ∈ ℕ.  | A000027
Triangular numbers t(n) | 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, ...  | t(n) = C(n \+ 1, 2) = ⁠n(n \+ 1)/2⁠ = 1 + 2 + ... + n for n ≥ 1, with t(0) = 0 (empty sum).  | A000217
Square numbers n2 | 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, ...  | n2 = n × n | A000290
Tetrahedral numbers T(n) | 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, ...  | T(n) is the sum of the first n triangular numbers, with T(0) = 0 (empty sum).  | A000292
Square pyramidal numbers  | 0, 1, 5, 14, 30, 55, 91, 140, 204, 285, ...  |  ⁠n(n \+ 1)(2n \+ 1)/6⁠ : The number of stacked spheres in a pyramid with a square base.  | A000330
Cube numbers n3 | 0, 1, 8, 27, 64, 125, 216, 343, 512, 729, ...  | n3 = n × n × n | A000578
Fifth powers | 0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, ...  | n5 | A000584
Star numbers  | 1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, ...  | Sn = 6n(n − 1) + 1.  | A003154
Stella octangula numbers  | 0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, 2651, 3444, 4381, ...  | Stella octangula numbers: n(2n2 − 1), with n ≥ 0.  | A007588

## Types of primes

Main article: List of prime numbers

Name | First elements | Short description | OEIS

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Mersenne prime exponents  | 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, ...  | Primes p such that 2p − 1 is prime.  | A000043
Mersenne primes | 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, ...  | 2p − 1 is prime, where p is a prime.  | A000668
Wagstaff primes | 3, 11, 43, 683, 2731, 43691, ...  | A prime number p of the form p=2q+13 where q is an odd prime.  | A000979
Wieferich primes | 1093, 3511  | Primes p satisfying 2p−1 ≡ 1 (mod p2).  | A001220
Sophie Germain primes  | 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, ...  | A prime number p such that 2p \+ 1 is also prime.  | A005384
Wilson primes | 5, 13, 563  | Primes p satisfying (p−1)! ≡ −1 (mod p2).  | A007540
Happy numbers | 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, ...  | The numbers whose trajectory under iteration of sum of squares of digits map includes 1.  | A007770
Factorial primes | 2, 3, 5, 7, 23, 719, 5039, 39916801, ...  | A prime number that is one less or one more than a factorial (all factorials > 1 are even).  | A088054
Wolstenholme primes | 16843, 2124679  | Primes p satisfying (2p−1p−1)≡1(modp4).  | A088164
Ramanujan primes | 2, 11, 17, 29, 41, 47, 59, 67, ...  | The nth Ramanujan prime is the least integer Rn for which π(x) − π(x/2) ≥ n, for all x ≥ Rn.  | A104272

## Base-dependent

Name | First elements | Short description | OEIS

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|

Aronson's sequence | 1, 4, 11, 16, 24, 29, 33, 35, 39, 45, ...  | "t" is the first, fourth, eleventh, ... letter in this sentence, not counting spaces or commas.  | A005224
Palindromic numbers | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, ...  | A number that remains the same when its digits are reversed.  | A002113
Permutable primes | 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, ...  | The numbers for which every permutation of digits is a prime.  | A003459
Harshad numbers in base 10  | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, ...  | A Harshad number in base 10 is an integer that is divisible by the sum of its digits (when written in base 10).  | A005349
Factorions | 1, 2, 145, 40585, ...  | A natural number that equals the sum of the factorials of its decimal digits.  | A014080
Circular primes | 2, 3, 5, 7, 11, 13, 17, 37, 79, 113, ...  | The numbers which remain prime under cyclic shifts of digits.  | A016114
Home prime | 1, 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, ...  | For n ≥ 2, a(n) is the prime that is finally reached when you start with n, concatenate its prime factors (A037276) and repeat until a prime is reached; a(n) = −1 if no prime is ever reached.  | A037274
Undulating numbers | 101, 121, 131, 141, 151, 161, 171, 181, 191, 202, ...  | A number that has the digit form ababab.  | A046075
Equidigital numbers | 1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 35, 37, 41, 43, 47, 49, 53, 59, 61, 64, ...  | A number that has the same number of digits as the number of digits in its prime factorization, including exponents but excluding exponents equal to 1.  | A046758
Extravagant numbers | 4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30, 33, 34, 36, 38, ...  | A number that has fewer digits than the number of digits in its prime factorization (including exponents).  | A046760
Pandigital numbers | 1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689, 1023457698, 1023457869, 1023457896, ...  | Numbers containing the digits 0–9 such that each digit appears exactly once.  | A050278

## References

* OEIS core sequences

## External links

* Index to OEIS

* v
* t
* e

Sequences and series

Integer
sequences|

| Basic|

* Arithmetic progression
* Geometric progression
* Harmonic progression
* Square number
* Cubic number
* Factorial
* Powers of two
* Powers of 10

|

Advanced (list)|

* Complete sequence
* Fibonacci numbers
* Figurate number
* Heptagonal number
* Hexagonal number
* Lucas number
* Pell number
* Pentagonal number
* Polygonal number
* Triangular number

Properties of sequences|

* Cauchy sequence
* Monotone sequence
* Periodic sequence

Properties of series|

* Convergent series
* Divergent series
* Conditional convergence
* Absolute convergence
* Uniform convergence
* Alternating series
* Telescoping series

Explicit series|

| Convergent|

* 1/2 − 1/4 + 1/8 − 1/16 + ⋯
* 1/2 + 1/4 + 1/8 + 1/16 + ⋯
* 1/4 + 1/16 + 1/64 + 1/256 + ⋯
* 1 + 1/2s\+ 1/3s \+ ... (Riemann zeta function)

|

Divergent|

* 1 + 1 + 1 + 1 + ⋯
* 1 + 2 + 3 + 4 + ⋯
* 1 + 2 + 4 + 8 + ⋯
* 1 − 1 + 1 − 1 + ⋯ (Grandi's series)
* Infinite arithmetic series
* 1 − 2 + 3 − 4 + ⋯
* 1 − 2 + 4 − 8 + ⋯
* 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
* 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
* 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)

Kinds of series|

* Taylor series
* Power series
* Formal power series
* Laurent series
* Puiseux series
* Dirichlet series
* Trigonometric series
* Fourier series
* Generating series

Hypergeometric
series|

* Generalized hypergeometric series
* Hypergeometric function of a matrix argument
* Lauricella hypergeometric series
* Modular hypergeometric series
* Riemann's differential equation
* Theta hypergeometric series

* Book
* Category

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Original source: https://en.wikipedia.org/wiki/List of integer sequences. Read more

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