In number theory, a factorion in a given number base 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

, we define the sum of the factorials of the digits, to be the following:

is the factorial and

-factorion if it is a fixed point., and thus are trivial factorions, and all other factorions are nontrivial factorions.

.

.

is a sociable factorion if it is a periodic point for a positive integer, and forms a cycle, and a amicable factorion.

are preperiodic points for any number, once again, making it a preperiodic point. This means also that there are a finite number of factorions and cycles.

function's persistence, and undefined if it never reaches a fixed point.

Factorions for SFDb

b = (m − 1)!

. Then:

• .

  .

b = m! − m + 1

. Then:

• .

Table of factorions and cycles of SFDb

.

See also

• Arithmetic dynamics

• Dudeney number

• Happy number

• Kaprekar's constant

• Kaprekar number

• Meertens number

• Narcissistic number

• Perfect digit-to-digit invariant

• Perfect digital invariant

• Sum-product number

References

External links

• Factorion at Wolfram MathWorld