Factorion

In number theory, a factorion in a given number base [math]\displaystyle{ b }[/math] 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 [math]\displaystyle{ n }[/math] be a natural number. For a base [math]\displaystyle{ b \gt 1 }[/math], we define the sum of the factorials of the digits of [math]\displaystyle{ n }[/math], [math]\displaystyle{ \operatorname{SFD}_b : \mathbb{N} \rightarrow \mathbb{N} }[/math], to be the following:

[math]\displaystyle{ \operatorname{SFD}_b(n) = \sum_{i=0}^{k - 1} d_i!. }[/math]

where [math]\displaystyle{ k = \lfloor \log_b n \rfloor + 1 }[/math] is the number of digits in the number in base [math]\displaystyle{ b }[/math], [math]\displaystyle{ n! }[/math] is the factorial of [math]\displaystyle{ n }[/math] and

[math]\displaystyle{ d_i = \frac{n \bmod{b^{i+1}} - n \bmod{b^{i}}}{b^{i}} }[/math]

is the value of the [math]\displaystyle{ i }[/math]th digit of the number. A natural number [math]\displaystyle{ n }[/math] is a [math]\displaystyle{ b }[/math]-factorion if it is a fixed point for [math]\displaystyle{ \operatorname{SFD}_b }[/math], i.e. if [math]\displaystyle{ \operatorname{SFD}_b(n) = n }[/math]. [math]\displaystyle{ 1 }[/math] and [math]\displaystyle{ 2 }[/math] are fixed points for all bases [math]\displaystyle{ b }[/math], and thus are trivial factorions for all [math]\displaystyle{ b }[/math], and all other factorions are nontrivial factorions.

For example, the number 145 in base [math]\displaystyle{ b = 10 }[/math] is a factorion because [math]\displaystyle{ 145 = 1! + 4! + 5! }[/math].

For [math]\displaystyle{ b = 2 }[/math], the sum of the factorials of the digits is simply the number of digits [math]\displaystyle{ k }[/math] in the base 2 representation since [math]\displaystyle{ 0! = 1! = 1 }[/math].

A natural number [math]\displaystyle{ n }[/math] is a sociable factorion if it is a periodic point for [math]\displaystyle{ \operatorname{SFD}_b }[/math], where [math]\displaystyle{ \operatorname{SFD}_b^k(n) = n }[/math] for a positive integer [math]\displaystyle{ k }[/math], and forms a cycle of period [math]\displaystyle{ k }[/math]. A factorion is a sociable factorion with [math]\displaystyle{ k = 1 }[/math], and a amicable factorion is a sociable factorion with [math]\displaystyle{ k = 2 }[/math].

All natural numbers [math]\displaystyle{ n }[/math] are preperiodic points for [math]\displaystyle{ \operatorname{SFD}_b }[/math], regardless of the base. This is because all natural numbers of base [math]\displaystyle{ b }[/math] with [math]\displaystyle{ k }[/math] digits satisfy [math]\displaystyle{ b^{k-1} \leq n \leq (b-1)!(k) }[/math]. However, when [math]\displaystyle{ k \geq b }[/math], then [math]\displaystyle{ b^{k-1} \gt (b-1)!(k) }[/math] for [math]\displaystyle{ b \gt 2 }[/math], so any [math]\displaystyle{ n }[/math] will satisfy [math]\displaystyle{ n \gt \operatorname{SFD}_b(n) }[/math] until [math]\displaystyle{ n \lt b^b }[/math]. There are finitely many natural numbers less than [math]\displaystyle{ b^b }[/math], so the number is guaranteed to reach a periodic point or a fixed point less than [math]\displaystyle{ b^b }[/math], making it a preperiodic point. For [math]\displaystyle{ b = 2 }[/math], the number of digits [math]\displaystyle{ k \leq n }[/math] 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 [math]\displaystyle{ b }[/math].

The number of iterations [math]\displaystyle{ i }[/math] needed for [math]\displaystyle{ \operatorname{SFD}_b^i(n) }[/math] to reach a fixed point is the [math]\displaystyle{ \operatorname{SFD}_b }[/math] function's persistence of [math]\displaystyle{ n }[/math], and undefined if it never reaches a fixed point.

Factorions for SFD_b

b = (k − 1)!

Let [math]\displaystyle{ k }[/math] be a positive integer and the number base [math]\displaystyle{ b = (k - 1)! }[/math]. Then:

• [math]\displaystyle{ n_1 = kb + 1 }[/math] is a factorion for [math]\displaystyle{ \operatorname{SFD}_b }[/math] for all [math]\displaystyle{ k. }[/math]

Proof

Let the digits of [math]\displaystyle{ n_1 = d_1 b + d_0 }[/math] be [math]\displaystyle{ d_1 = k }[/math], and [math]\displaystyle{ d_0 = 1. }[/math] Then

[math]\displaystyle{ \operatorname{SFD}_b(n_1) = d_1! + d_0! }[/math]

[math]\displaystyle{ = k! + 1! }[/math]

[math]\displaystyle{ = k(k - 1)! + 1 }[/math]

[math]\displaystyle{ = d_1 b + d_0 }[/math]

[math]\displaystyle{ = n_1 }[/math]

Thus [math]\displaystyle{ n_1 }[/math] is a factorion for [math]\displaystyle{ F_b }[/math] for all [math]\displaystyle{ k }[/math].

• [math]\displaystyle{ n_2 = kb + 2 }[/math] is a factorion for [math]\displaystyle{ \operatorname{SFD}_b }[/math] for all [math]\displaystyle{ k }[/math].

Proof

Let the digits of [math]\displaystyle{ n_2 = d_1 b + d_0 }[/math] be [math]\displaystyle{ d_1 = k }[/math], and [math]\displaystyle{ d_0 = 2 }[/math]. Then

[math]\displaystyle{ \operatorname{SFD}_b(n_2) = d_1! + d_0! }[/math]

[math]\displaystyle{ = k! + 2! }[/math]

[math]\displaystyle{ = k(k - 1)! + 2 }[/math]

[math]\displaystyle{ = d_1 b + d_0 }[/math]

[math]\displaystyle{ = n_2 }[/math]

Thus [math]\displaystyle{ n_2 }[/math] is a factorion for [math]\displaystyle{ F_b }[/math] for all [math]\displaystyle{ k }[/math].

Factorions

[math]\displaystyle{ k }[/math]	[math]\displaystyle{ b }[/math]	[math]\displaystyle{ n_1 }[/math]	[math]\displaystyle{ n_2 }[/math]
4	6	41	42
5	24	51	52
6	120	61	62
7	720	71	72

b = k! − k + 1

Let [math]\displaystyle{ k }[/math] be a positive integer and the number base [math]\displaystyle{ b = k! - k + 1 }[/math]. Then:

• [math]\displaystyle{ n_1 = b + k }[/math] is a factorion for [math]\displaystyle{ \operatorname{SFD}_b }[/math] for all [math]\displaystyle{ k }[/math].

Proof

Let the digits of [math]\displaystyle{ n_1 = d_1 b + d_0 }[/math] be [math]\displaystyle{ d_1 = 1 }[/math], and [math]\displaystyle{ d_0 = k }[/math]. Then

[math]\displaystyle{ \operatorname{SFD}_b(n_1) = d_1! + d_0! }[/math]

[math]\displaystyle{ = 1! + k! }[/math]

[math]\displaystyle{ = k! + 1 - k + k }[/math]

[math]\displaystyle{ = 1(k! - k + 1) + k }[/math]

[math]\displaystyle{ = d_1 b + d_0 }[/math]

[math]\displaystyle{ = n_1 }[/math]

Thus [math]\displaystyle{ n_1 }[/math] is a factorion for [math]\displaystyle{ F_b }[/math] for all [math]\displaystyle{ k }[/math].

Factorions

[math]\displaystyle{ k }[/math]	[math]\displaystyle{ b }[/math]	[math]\displaystyle{ n_1 }[/math]
3	4	13
4	21	14
5	116	15
6	715	16

Table of factorions and cycles of SFD_b

All numbers are represented in base [math]\displaystyle{ b }[/math].

Base [math]\displaystyle{ b }[/math]	Nontrivial factorion ([math]\displaystyle{ n \neq 1 }[/math], [math]\displaystyle{ n \neq 2 }[/math])	Cycles
2	[math]\displaystyle{ \varnothing }[/math]	[math]\displaystyle{ \varnothing }[/math]
3	[math]\displaystyle{ \varnothing }[/math]	[math]\displaystyle{ \varnothing }[/math]
4	13	3 → 12 → 3
5	144	[math]\displaystyle{ \varnothing }[/math]
6	41, 42	[math]\displaystyle{ \varnothing }[/math]
7	[math]\displaystyle{ \varnothing }[/math]	36 → 2055 → 465 → 2343 → 53 → 240 → 36
8	[math]\displaystyle{ \varnothing }[/math]	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