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.

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 $$k.$$
 * $$n_2 = kb + 2$$ is a factorion for $$\operatorname{SFD}_b$$ for all $$k$$.

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$$.

Table of factorions and cycles of $SFD_{b}$
All numbers are represented in base $$b$$.