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 $n$, $\operatorname {SFD} _{b}:\mathbb {N} \rightarrow \mathbb {N}$, 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 $i$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 $\operatorname {SFD} _{b}^{c}(n)=n$ 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 $b^{k-1}\leq n<b^{k}$. 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 $n<b^{b}$. 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 $\operatorname {SFD} _{b}^{i}(n)$ 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 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 $m\geq 4$.

Proof

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 $m\geq 4$.

Proof

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 $k$.

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 $\operatorname {SFD} _{b}$ for all $m\geq 3$.

Proof

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 $m$.

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 ($n\neq 1$, $n\neq 2$)	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