Factorion

In number theory, a factorion in a given number base $b$ 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 $n$, ${\mathrm{SFD}}_b:\mathbb{N}\to \mathbb{N}$, to be the following:

${\mathrm{SFD}}_b(n)=\sum _{i=0}^{k-1}{d}_i!.$

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

$d_i=\frac{nmod{b}^{i+1}-nmod{b}^i}{b^i}$

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 ${\mathrm{SFD}}_b$, i.e. if ${\mathrm{SFD}}_b(n)=n$. $1$ 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 ${\mathrm{SFD}}_b$, where ${\mathrm{SFD}}_b^c(n)=n$ 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 ${\mathrm{SFD}}_b$, regardless of the base. This is because all natural numbers of base $b$ with $k$ digits satisfy $b^{k-1}\le n<b^k$. Given that each of the $k$ digits is at most $b-1$, ${\mathrm{SFD}}_b\le (b-1)!k$. However, when $k\ge b$, then $b^{k-1}>(b-1)!(k)$ for $b>2$, so any $n$ will satisfy $n>{\mathrm{SFD}}_b(n)$ 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 $k\le n$ 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 ${\mathrm{SFD}}_b^i(n)$ to reach a fixed point is the ${\mathrm{SFD}}_b$ function's persistence of $n$, and undefined if it never reaches a fixed point.

Factorions for 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 ${\mathrm{SFD}}_b$ for all $m\ge 4$.

Proof

Let the digits of $n_1=d_{1}b+d_0$ be $d_1=m$, and $d_0=1.$ Then

${\mathrm{SFD}}_b(n_1)=d_1!+d_0!$

$=m!+1!$

$=m(m-1)!+1$

$=d_{1}b+d_0$

$=n_1$

Thus $n_1$ is a factorion for $F_b$ for all $k$.

• $n_2=mb+2$ is a factorion for ${\mathrm{SFD}}_b$ for all $m\ge 4$.

Proof

Let the digits of $n_2=d_{1}b+d_0$ be $d_1=m$, and $d_0=2$. Then

${\mathrm{SFD}}_b(n_2)=d_1!+d_0!$

$=m!+2!$

$=m(m-1)!+2$

$=d_{1}b+d_0$

$=n_2$

Thus $n_2$ is a factorion for $F_b$ for all $k$.

Factorions

$m$	$b$	$n_1$	$n_2$
4	6	41	42
5	24	51	52
6	120	61	62
7	720	71	72

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 ${\mathrm{SFD}}_b$ for all $m\ge 3$.

Proof

Let the digits of $n_1=d_{1}b+d_0$ be $d_1=1$, and $d_0=m$. Then

${\mathrm{SFD}}_b(n_1)=d_1!+d_0!$

$=1!+m!$

$=m!+1-m+m$

$=1(m!-m+1)+m$

$=d_{1}b+d_0$

$=n_1$

Thus $n_1$ is a factorion for $F_b$ for all $m$.

Factorions

$m$	$b$	$n_1$
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 $b$.

Base $b$	Nontrivial factorion ($n\ne 1$, $n\ne 2$)	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