Factorion

Definition

Let n be a natural number. For a base b 1, we define the sum of the factorials of the digits of n, \operatorname{SFD}_b : \mathbb{N} \rightarrow \mathbb{N}, to be the following: :\operatorname{SFD}b(n) = \sum{i=0}^{k - 1} d_i!. where k = \lfloor \log_b n \rfloor + 1 is the number of digits in the number in base b, n! is the factorial of n and :d_i = \frac{n \bmod{b^{i+1}} - n \bmod{b^{i}}}{b^{i}} 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 \operatorname{SFD}_b, i.e. if \operatorname{SFD}_b(n) = n. 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 145 = 1! + 4! + 5!.

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 \operatorname{SFD}_b, where \operatorname{SFD}_b^c(n) = n for a positive integer c, and forms a cycle of period c. A factorion is a sociable factorion with c = 1, and a amicable factorion is a sociable factorion with c = 2.

All natural numbers n are preperiodic points for \operatorname{SFD}_b, regardless of the base. This is because all natural numbers of base b with k digits satisfy b^{k-1} \leq n . Given that each of the k digits is at most b-1, \operatorname{SFD}_b \leq (b-1)!k. However, when k \geq b, then b^{k-1} (b-1)!(k) for b 2, so any n will satisfy n \operatorname{SFD}_b(n) until n . 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 \operatorname{SFD}_b^i(n) to reach a fixed point is the \operatorname{SFD}_b function's persistence of n, and undefined if it never reaches a fixed point.

Factorions for {{Math|{{math|SFD_''b''}}}}

''b'' = (''m'' − 1)!

Let m be a positive integer and the number base b = (m - 1)!. Then:

• n_1 = mb + 1 is a factorion for \operatorname{SFD}_b for all m\geq 4. Let the digits of n_1 = d_1 b + d_0 be d_1 = m, and d_0 = 1. Then : \operatorname{SFD}_b(n_1) = d_1! + d_0! :: = m! + 1! :: = m(m - 1)! + 1 :: = d_1 b + d_0 :: = n_1 Thus n_1 is a factorion for F_b for all k.
• n_2 = mb + 2 is a factorion for \operatorname{SFD}_b for all m\geq 4. Let the digits of n_2 = d_1 b + d_0 be d_1 = m, and d_0 = 2. Then : \operatorname{SFD}_b(n_2) = d_1! + d_0! :: = m! + 2! :: = m(m - 1)! + 2 :: = d_1 b + d_0 :: = n_2 Thus n_2 is a factorion for F_b for all k.

''b'' = ''m''! − ''m'' + 1

Let k be a positive integer and the number base b = m! - m + 1. Then:

• n_1 = b + m is a factorion for \operatorname{SFD}_b for all m\geq 3. Let the digits of n_1 = d_1 b + d_0 be d_1 = 1, and d_0 = m. Then : \operatorname{SFD}_b(n_1) = d_1! + d_0! :: = 1! + m! :: = m! + 1 - m + m :: = 1(m! - m + 1) + m :: = d_1 b + d_0 :: = n_1 Thus n_1 is a factorion for F_b for all m.

Table of factorions and cycles of {{Math|{{math|SFD_''b''}}}}

All numbers are represented in base b.

References

References

1. Sloane, Neil. "A014080".
2. Gardner, Martin. (1978). "Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-Of-Mind". Vintage Books.
3. Madachy, Joseph S.. (1979). "Madachy's Mathematical Recreations". Dover Publications.
4. Pickover, Clifford A.. (1995). "Keys to Infinity". John Wiley & Sons.
5. Gupta, Shyam S.. (2004). "Sum of the Factorials of the Digits of Integers". The Mathematical Association.
6. Sloane, Neil. "A061602".
7. Abbott, Steve. (2004). "SFD Chains and Factorion Cycles". The Mathematical Association.
8. Sloane, Neil. "A214285".
9. Sloane, Neil. "A254499".
10. Sloane, Neil. "A193163".