Factorion

In number theory, a factorion in a given number base b is a natural number that equals the sum of the factorials of its digits.[1][2][3] The name factorion was coined by the author Clifford A. Pickover.[4]

## Contents

* 1 Definition
* 2 Factorions for SFDb
* 2.1 b = (m − 1)!
* 2.2 b = m! − m \+ 1
* 2.3 Table of factorions and cycles of SFDb
* 3 See also
* 4 References
* 5 External links

## Definition

Let n be a natural number. For a base b>1, we define the sum of the factorials of the digits[5][6] of n, SFDb:ℕ->ℕ, to be the following:

SFDb(n)=∑i=0k−1di!.

where k=⌊logbn⌋+1 is the number of digits in the number in base b, n! is the factorial of n and

di=nmodbi+1−nmodbibi

is the value of the ith digit of the number. A natural number n is a b-factorion if it is a fixed point for SFDb, i.e. if SFDb(n)=n.[7] 1 and 2 are fixed points for all bases b, and thus are trivial factorions for all b, and all other factorions are nontrivial factorions.

For example, the number 145 in base b=10 is a factorion because 145=1!+4!+5!.

For b=2, the sum of the factorials of the digits is simply the number of digits k in the base 2 representation since 0!=1!=1.

A natural number n is a sociable factorion if it is a periodic point for SFDb, where SFDbc(n)=n for a positive integer c, and forms a cycle of period c. A factorion is a sociable factorion with c=1, and a amicable factorion is a sociable factorion with c=2.[8][9]

All natural numbers n are preperiodic points for SFDb, regardless of the base. This is because all natural numbers of base b with k digits satisfy bk−1≤n<bk. Given that each of the k digits is at most b−1, SFDb≤(b−1)!k. However, when k≥b, then bk−1>(b−1)!(k) for b>2, so any n will satisfy n>SFDb(n) until n<bb. There are finitely many natural numbers less than bb, so the number is guaranteed to reach a periodic point or a fixed point less than bb, making it a preperiodic point. For b=2, the number of digits k≤n for any number, once again, making it a preperiodic point. This means also that there are a finite number of factorions and cycles for any given base b.

The number of iterations i needed for SFDbi(n) to reach a fixed point is the SFDb function's persistence of n, and undefined if it never reaches a fixed point.

## Factorions for SFDb

### b = (m − 1)!

Let m be a positive integer and the number base b=(m−1)!. Then:

* n1=mb+1 is a factorion for SFDb for all m≥4.

Proof

Let the digits of n1=d1b+d0 be d1=m, and d0=1. Then

SFDb(n1)=d1!+d0!

=m!+1!
=m(m−1)!+1
=d1b+d0
=n1

Thus n1 is a factorion for Fb for all k.

* n2=mb+2 is a factorion for SFDb for all m≥4.

Proof

Let the digits of n2=d1b+d0 be d1=m, and d0=2. Then

SFDb(n2)=d1!+d0!

=m!+2!
=m(m−1)!+2
=d1b+d0
=n2

Thus n2 is a factorion for Fb for all k.

Factorions  m | b | n1 | n2

|
|
|

4 | 6 | 41 | 42
5 | 24 | 51 | 52
6 | 120 | 61 | 62
7 | 720 | 71 | 72

### b = m! − m \+ 1

Let k be a positive integer and the number base b=m!−m+1. Then:

* n1=b+m is a factorion for SFDb for all m≥3.

Proof

Let the digits of n1=d1b+d0 be d1=1, and d0=m. Then

SFDb(n1)=d1!+d0!

=1!+m!
=m!+1−m+m
=1(m!−m+1)+m
=d1b+d0
=n1

Thus n1 is a factorion for Fb for all m.

Factorions  m | b | n1

|
|

3 | 4 | 13
4 | 21 | 14
5 | 116 | 15
6 | 715 | 16

### Table of factorions and cycles of SFDb

All numbers are represented in base b.

Base b | Nontrivial factorion (n≠1, n≠2)[10] | Cycles

|
|

2 | ∅ | ∅
3 | ∅ | ∅
4 | 13 | 3 → 12 → 3
5 | 144 | ∅
6 | 41, 42 | ∅
7 | ∅ | 36 → 2055 → 465 → 2343 → 53 → 240 → 36
8 | ∅ |

3 → 6 → 1320 → 12

175 → 12051 → 175

9 | 62558 |
10 | 145, 40585 |

871 → 45361 → 871[9]

872 → 45362 → 872[8]

169 → 363601 → 1454 → 169[11]

## See also

* Arithmetic dynamics
* Dudeney number
* Happy number
* Kaprekar's constant
* Kaprekar number
* Meertens number
* Narcissistic number
* Perfect digit-to-digit invariant
* Perfect digital invariant
* Sum-product number

## References

1. ↑ Sloane, Neil, A014080, https://oeis.org/A014080
2. ↑ Gardner, Martin (1978), "Factorial Oddities", Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-Of-Mind, Vintage Books, pp. 61 and 64, ISBN 9780394726236, https://books.google.com/books?id=RcnbvQEACAAJ&q=Mathematical+Magic+Show:+More+Puzzles,+Games,+Diversions,+Illusions+and+Other+Mathematical+Sleight-Of-Mind
3. ↑ Madachy, Joseph S. (1979), Madachy's Mathematical Recreations, Dover Publications, p. 167, ISBN 9780486237626, https://books.google.com/books?id=UvpUAAAAYAAJ&q=Madachy's+Mathematical+Recreations
4. ↑ Pickover, Clifford A. (1995), "The Loneliness of the Factorions", Keys to Infinity, John Wiley & Sons, pp. 169–171 and 319–320, ISBN 9780471193340, https://books.google.com/books?id=oXIFAAAACAAJ&q=Keys+to+Infinity
5. ↑ Gupta, Shyam S. (2004), "Sum of the Factorials of the Digits of Integers", The Mathematical Gazette (The Mathematical Association) 88 (512): 258–261, doi:10.1017/S0025557200174996
6. ↑ Sloane, Neil, A061602, https://oeis.org/A061602
7. ↑ Abbott, Steve (2004), "SFD Chains and Factorion Cycles", The Mathematical Gazette (The Mathematical Association) 88 (512): 261–263, doi:10.1017/S002555720017500X
8. ↑ 8.0 8.1 Sloane, Neil, A214285, https://oeis.org/A214285
9. ↑ 9.0 9.1 Sloane, Neil, A254499, https://oeis.org/A254499
10. ↑ Sloane, Neil, A193163, https://oeis.org/A193163
11. ↑ Sloane, Neil. "A188284". https://oeis.org/A188284.

## External links

* Factorion at Wolfram MathWorld

* v
* t
* e

Classes of natural numbers

Powers and
related numbers|

* Achilles
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Of the form
a × 2b ± 1|

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Other polynomial
numbers|

* Carol
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Recursively defined
numbers|

* Fibonacci
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Possessing a
specific set
of other numbers|

* Knödel
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Expressible via
specific sums|

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Figurate numbers|

| 2-dimensional|

| centered|

* Centered triangular
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|

non-centered|

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3-dimensional|

| centered|

* Centered tetrahedral
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|

non-centered|

* Tetrahedral
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pyramidal|

* Square pyramidal
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4-dimensional|

| centered|

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|

non-centered|

* Pentatope
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Base-dependent
numbers|

| Arithmetic functions and dynamics|

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|

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numbers|

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Pseudoprimes|

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Arithmetic functions|

| By properties of σ(n)|

* Abundant
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* Untouchable

|

By properties of Ω(n)|

* Almost prime
* Semiprime

By properties of φ(n)|

* Highly cototient
* Highly totient
* Noncototient
* Nontotient
* Perfect totient
* Sparsely totient

By properties of s(n)|

* Almost perfect
* Amicable
* Betrothed
* Deficient
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* Sociable

Other|

* Euclid
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Dividing a quotient|

* Wieferich
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Other prime factor
or divisor related
numbers|

* Blum
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via a sieve|

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