Английская Википедия:Factorion
In number theory, a factorion in a given number base <math>b</math> 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]
Definition
Let <math>n</math> be a natural number. For a base <math>b > 1</math>, we define the sum of the factorials of the digits[5][6] of <math>n</math>, <math>\operatorname{SFD}_b : \mathbb{N} \rightarrow \mathbb{N}</math>, to be the following:
- <math>\operatorname{SFD}_b(n) = \sum_{i=0}^{k - 1} d_i!.</math>
where <math>k = \lfloor \log_b n \rfloor + 1</math> is the number of digits in the number in base <math>b</math>, <math>n!</math> is the factorial of <math>n</math> and
- <math>d_i = \frac{n \bmod{b^{i+1}} - n \bmod{b^{i}}}{b^{i}}</math>
is the value of the <math>i</math>th digit of the number. A natural number <math>n</math> is a <math>b</math>-factorion if it is a fixed point for <math>\operatorname{SFD}_b</math>, i.e. if <math>\operatorname{SFD}_b(n) = n</math>.[7] <math>1</math> and <math>2</math> are fixed points for all bases <math>b</math>, and thus are trivial factorions for all <math>b</math>, and all other factorions are nontrivial factorions.
For example, the number 145 in base <math>b = 10</math> is a factorion because <math>145 = 1! + 4! + 5!</math>.
For <math>b = 2</math>, the sum of the factorials of the digits is simply the number of digits <math>k</math> in the base 2 representation since <math>0! = 1! = 1</math>.
A natural number <math>n</math> is a sociable factorion if it is a periodic point for <math>\operatorname{SFD}_b</math>, where <math>\operatorname{SFD}_b^k(n) = n</math> for a positive integer <math>k</math>, and forms a cycle of period <math>k</math>. A factorion is a sociable factorion with <math>k = 1</math>, and a amicable factorion is a sociable factorion with <math>k = 2</math>.[8][9]
All natural numbers <math>n</math> are preperiodic points for <math>\operatorname{SFD}_b</math>, regardless of the base. This is because all natural numbers of base <math>b</math> with <math>k</math> digits satisfy <math>b^{k-1} \leq n \leq (b-1)!(k)</math>. However, when <math>k \geq b</math>, then <math>b^{k-1} > (b-1)!(k)</math> for <math>b > 2</math>, so any <math>n</math> will satisfy <math>n > \operatorname{SFD}_b(n)</math> until <math>n < b^b</math>. There are finitely many natural numbers less than <math>b^b</math>, so the number is guaranteed to reach a periodic point or a fixed point less than <math> b^b</math>, making it a preperiodic point. For <math>b = 2</math>, the number of digits <math>k \leq n</math> 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 <math>b</math>.
The number of iterations <math>i</math> needed for <math>\operatorname{SFD}_b^i(n)</math> to reach a fixed point is the <math>\operatorname{SFD}_b</math> function's persistence of <math>n</math>, and undefined if it never reaches a fixed point.
Factorions for Шаблон:Math
b = (k − 1)!
Let <math>k</math> be a positive integer and the number base <math>b = (k - 1)!</math>. Then:
- <math>n_1 = kb + 1</math> is a factorion for <math>\operatorname{SFD}_b</math> for all <math>k.</math>
- <math>n_2 = kb + 2</math> is a factorion for <math>\operatorname{SFD}_b</math> for all <math>k</math>.
| <math>k</math> | <math>b</math> | <math>n_1</math> | <math>n_2</math> |
|---|---|---|---|
| 4 | 6 | 41 | 42 |
| 5 | 24 | 51 | 52 |
| 6 | 120 | 61 | 62 |
| 7 | 720 | 71 | 72 |
b = k! − k + 1
Let <math>k</math> be a positive integer and the number base <math>b = k! - k + 1</math>. Then:
- <math>n_1 = b + k</math> is a factorion for <math>\operatorname{SFD}_b</math> for all <math>k</math>.
| <math>k</math> | <math>b</math> | <math>n_1</math> |
|---|---|---|
| 3 | 4 | 13 |
| 4 | 21 | 14 |
| 5 | 116 | 15 |
| 6 | 715 | 16 |
Table of factorions and cycles of Шаблон:Math
All numbers are represented in base <math>b</math>.
| Base <math>b</math> | Nontrivial factorion (<math>n \neq 1</math>, <math>n \neq 2</math>)[10] | Cycles |
|---|---|---|
| 2 | <math>\varnothing</math> | <math>\varnothing</math> |
| 3 | <math>\varnothing</math> | <math>\varnothing</math> |
| 4 | 13 | 3 → 12 → 3 |
| 5 | 144 | <math>\varnothing</math> |
| 6 | 41, 42 | <math>\varnothing</math> |
| 7 | <math>\varnothing</math> | 36 → 2055 → 465 → 2343 → 53 → 240 → 36 |
| 8 | <math>\varnothing</math> |
3 → 6 → 1320 → 12 175 → 12051 → 175 |
| 9 | 62558 | |
| 10 | 145, 40585 |
871 → 45361 → 871[9] 872 → 45362 → 872[8] |
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
External links
Шаблон:Classes of natural numbers
