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,, to be the following: where is the number of digits in the number in base b, n! is the factorial of n and 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, i.e. if. 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. 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, where 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, regardless of the base. This is because all natural numbers of base b with k digits satisfy. However, when k \geq b, then for b > 2, so any n will satisfy 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 to reach a fixed point is the function's persistence of n, and undefined if it never reaches a fixed point.

Factorions for

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b = (k − 1)!

Let k be a positive integer and the number base. Then:

b = k! − k + 1

Let k be a positive integer and the number base. Then:

Table of factorions and cycles of

{{math|SFD{{sub|b}}}} All numbers are represented in base b.