# 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. The name factorion was coined by the author Clifford A. Pickover.

## Definition

Let $n$ be a natural number. For a base $b>1$, we define the sum of the factorials of the digits of $n$, $\operatorname {SFD} _{b}:\mathbb {N} \rightarrow \mathbb {N}$, to be the following:
    $\operatorname {SFD} _{b}(n)=\sum _{i=0}^{k-1}d_{i}!.$
where $k=\lfloor \log _{b}n\rfloor +1$ is the number of digits in the number in base $b$, $n!$ is the factorial of $n$ and
    $d_{i}={\frac {n{\bmod {b^{i+1}}}-n{\bmod {b^{i}}}}{b^{i}}}$
is the value of the $i$th digit of the number. A natural number $n$ is a $b$-factorion if it is a fixed point for $\operatorname {SFD} _{b}$, i.e. if $\operatorname {SFD} _{b}(n)=n$. $1$ 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 $\operatorname {SFD} _{b}$, where $\operatorname {SFD} _{b}^{k}(n)=n$ for a positive integer $k$, and forms a cycle of period $k$. A factorion is a sociable factorion with $k=1$, and a amicable factorion is a sociable factorion with $k=2$.
All natural numbers $n$ are preperiodic points for $\operatorname {SFD} _{b}$, regardless of the base. This is because all natural numbers of base $b$ with $k$ digits satisfy $b^{k-1}\leq n\leq (b-1)!(k)$. However, when $k\geq b$, then $b^{k-1}>(b-1)!(k)$ for $b>2$, so any $n$ will satisfy $n>\operatorname {SFD} _{b}(n)$ until $n<b^{b}$. There are finitely many natural numbers less than $b^{b}$, so the number is guaranteed to reach a periodic point or a fixed point less than $b^{b}$, making it a preperiodic point. For $b=2$, the number of digits $k\leq n$ 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 $\operatorname {SFD} _{b}^{i}(n)$ to reach a fixed point is the $\operatorname {SFD} _{b}$ function's persistence of $n$, and undefined if it never reaches a fixed point.

## Factorions for SFD$_b$
### b = (k − 1)!

Let $k$ be a positive integer and the number base $b=(k-1)!$. Then:
* $n_{1}=kb+1$ is a factorion for $\operatorname {SFD} _{b}$ for all $k.$
Let the digits of $n_{1}=d_{1}b+d_{0}$ be $d_{1}=k$, and $d_{0}=1.$ Then
    $\operatorname {SFD} _{b}(n_{1})=d_{1}!+d_{0}!$
       $=k!+1!$
       $=k(k-1)!+1$
       $=d_{1}b+d_{0}$
       $=n_{1}$
Thus $n_{1}$ is a factorion for $F_{b}$ for all $k$.
* $n_{2}=kb+2$ is a factorion for $\operatorname {SFD} _{b}$ for all $k$.
Let the digits of $n_{2}=d_{1}b+d_{0}$ be $d_{1}=k$, and $d_{0}=2$. Then
    $\operatorname {SFD} _{b}(n_{2})=d_{1}!+d_{0}!$
       $=k!+2!$
       $=k(k-1)!+2$
       $=d_{1}b+d_{0}$
       $=n_{2}$
Thus $n_{2}$ is a factorion for $F_{b}$ for all $k$.
Factorions :
| $k$ | $b$ | $n_{1}$ | $n_{2}$ |
| 4 | 6 | 41 | 42 |
| 5 | 24 | 51 | 52 |
| 6 | 120 | 61 | 62 |
| 7 | 720 | 71 | 72 |

### b = k! − k + 1

Let $k$ be a positive integer and the number base $b=k!-k+1$. Then:
* $n_{1}=b+k$ is a factorion for $\operatorname {SFD} _{b}$ for all $k$.
Let the digits of $n_{1}=d_{1}b+d_{0}$ be $d_{1}=1$, and $d_{0}=k$. Then
    $\operatorname {SFD} _{b}(n_{1})=d_{1}!+d_{0}!$
       $=1!+k!$
       $=k!+1-k+k$
       $=1(k!-k+1)+k$
       $=d_{1}b+d_{0}$
       $=n_{1}$
Thus $n_{1}$ is a factorion for $F_{b}$ for all $k$.
Factorions :
| $k$ | $b$ | $n_{1}$ |
| 3 | 4 | 13 |
| 4 | 21 | 14 |
| 5 | 116 | 15 |
| 6 | 715 | 16 |

### Table of factorions and cycles of SFD$_b$

All numbers are represented in base $b$.
| Base $b$ | Nontrivial factorion ($n\neq 1$, $n\neq 2$) | Cycles |
| 2 | $\varnothing$ | $\varnothing$ |
| 3 | $\varnothing$ | $\varnothing$ |
| 4 | 13 | 3 → 12 → 3 |
| 5 | 144 | $\varnothing$ |
| 6 | 41, 42 | $\varnothing$ |
| 7 | $\varnothing$ | 36 → 2055 → 465 → 2343 → 53 → 240 → 36 |
| 8 | $\varnothing$ | 3 → 6 → 1320 → 12 175 → 12051 → 175 |
| 9 | 62558 | |
| 10 | 145, 40585 | 871 → 45361 → 871 872 → 45362 → 872 |