Factorion Explained

b 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

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} : N → N , to be the following:\operatorname{SFD} _{b(n)} = \sum ^{k -1} _{i =0} 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} = (n \bmod{b ^{i+1})/(-) n \bmod{b ^{i} }} is the value of the i th 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} ^{k(n)} _{b} = n for a positive integer k , and forms a cycle of period k . A factorion is a sociable factorion with k = 1 , and a amicable factorion is a sociable factorion with k = 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 \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 < b ^{b} . 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} ^{i(n)} _{b} 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

b = (k − 1)!

Let

k be a positive integer and the number base b = (k - 1 )! . Then:n _{1} = kb + 1 is a factorion for \operatorname{SFD} _{b} for all k. n _{2} = kb + 2 is a factorion for \operatorname{SFD} _{b} for all k .

Factorions! k ! b ! n _{1} ! n _{2}

4		41	42
5	24	51	52
6	120	61	62
7	720	71	72

b = k! − k + 1

Let

k be a positive integer and the number base b = k! - k + 1 . Then:n _{1} = b + k is a factorion for \operatorname{SFD} _{b} for all k .

Factorions! k ! b ! n _{1}

3		13
4	21	14
5	116	15
6	715	16

Table of factorions and cycles of

All numbers are represented in base

b .

	Base b		Nontrivial factorion (n ≠ 1 , n ≠ 2 )		Cycles
2	\varnothing	\varnothing
3	\varnothing	\varnothing
4	13	3 → 12 → 3
5	144	\varnothing
6	41, 42	\varnothing
7	\varnothing	36 → 2055 → 465 → 2343 → 53 → 240 → 36
8	\varnothing	3 → 6 → 1320 → 12 175 → 12051 → 175
9	62558
10	145, 40585	871 → 45361 → 871 872 → 45362 → 872

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

External links

• Factorion at Wolfram MathWorld