In number theory, a factorion in a given number base ${\displaystyle b}$ is a natural number that equals the sum of the factorials of its digits.^[1][2][3] The name factorion was coined by the author Clifford A. Pickover.^[4]

Let ${\displaystyle n}$ be a natural number. For a base ${\displaystyle b>1}$, we define the sum of the factorials of the digits^[5][6] of ${\displaystyle n}$, ${\displaystyle \operatorname {SFD} _{b}:\mathbb {N} \rightarrow \mathbb {N} }$, to be the following:

${\displaystyle \operatorname {SFD} _{b}(n)=\sum _{i=0}^{k-1}d_{i}!.}$

where ${\displaystyle k=\lfloor \log _{b}n\rfloor +1}$ is the number of digits in the number in base ${\displaystyle b}$, ${\displaystyle n!}$ is the factorial of ${\displaystyle n}$ and

${\displaystyle d_{i}={\frac {n{\bmod {b^{i+1}}}-n{\bmod {b^{i}}}}{b^{i}}}}$

is the value of the ${\displaystyle i}$th digit of the number. A natural number ${\displaystyle n}$ is a ${\displaystyle b}$-factorion if it is a fixed point for ${\displaystyle \operatorname {SFD} _{b}}$, i.e. if ${\displaystyle \operatorname {SFD} _{b}(n)=n}$.^[7] ${\displaystyle 1}$ and ${\displaystyle 2}$ are fixed points for all bases ${\displaystyle b}$, and thus are trivial factorions for all ${\displaystyle b}$, and all other factorions are nontrivial factorions.

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

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

A natural number ${\displaystyle n}$ is a sociable factorion if it is a periodic point for ${\displaystyle \operatorname {SFD} _{b}}$, where ${\displaystyle \operatorname {SFD} _{b}^{k}(n)=n}$ for a positive integer ${\displaystyle k}$, and forms a cycle of period ${\displaystyle k}$. A factorion is a sociable factorion with ${\displaystyle k=1}$, and a amicable factorion is a sociable factorion with ${\displaystyle k=2}$.^[8][9]

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

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

• 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