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. 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.
| 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.
| 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 |