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 Let be a natural number. For a base , we define the sum of the factorials of the digits of , , to be the following:
where is the number of digits in the number in base , is the factorial of and
is the value of the th digit of the number. A natural number is a -factorion if it is a fixed point for , i.e. if . and are fixed points for all bases , and thus are trivial factorions for all , and all other factorions are nontrivial factorions.
For example, the number 145 in base is a factorion because .
For , 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 , where for a positive integer , and forms a cycle of period . A factorion is a sociable factorion with , and a amicable factorion is a sociable factorion with .
All natural numbers are preperiodic points for , regardless of the base. This is because all natural numbers of base with digits satisfy . Given that each of the digits is at most , . However, when , then for , so any will satisfy until . 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. For , 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 needed for to reach a fixed point is the function's persistence of , and undefined if it never reaches a fixed point.
Factorions for b = (m − 1)! Let be a positive integer and the number base . Then: is a factorion for for all .
is a factorion for for all .
+ Factorions 4 6 41 42 5 24 51 52 6 120 61 62 7 720 71 72
b = m! − m + 1 Let be a positive integer and the number base . Then: is a factorion for for all .
+ Factorions 3 4 13 4 21 14 5 116 15 6 715 16
Table of factorions and cycles of All numbers are represented in base .
Base Nontrivial factorion (, ) Cycles 2 3 4 13 3 → 12 → 3 5 144 6 41, 42 7 36 → 2055 → 465 → 2343 → 53 → 240 → 36 8 3 → 6 → 1320 → 12
175 → 12051 → 175 9 62558 10 145, 40585 871 → 45361 → 871
872 → 45362 → 872
169 → 363601 → 1454 → 169
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