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 ith 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.