Factorion

In number theory, a factorion in a given number base

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.

be a natural number.

, we define the sum of the factorials of the digits[5][6] of

th digit of the number.

A natural number

are fixed points for all bases

For example, the number 145 in base

, the sum of the factorials of the digits is simply the number of digits

in the base 2 representation since

A natural number

is a sociable factorion if it is a periodic point for

for a positive integer

, and forms a cycle of period

[8][9] All natural numbers

This is because all natural numbers of base

digits satisfy

There are finitely many natural numbers less than

, so the number is guaranteed to reach a periodic point or a fixed point less than

, making it a preperiodic point.

, the number of digits

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

The number of iterations

to reach a fixed point is the

, and undefined if it never reaches a fixed point.

be a positive integer and the number base

be a positive integer and the number base

All numbers are represented in base