Factorion

Let ${\displaystyle n}$ be a natural number. We define the sum of the factorial of the digits[5][6] of ${\displaystyle n}$ for base ${\displaystyle b>1}$ ${\displaystyle SFD_{b}:\mathbb {N} \rightarrow \mathbb {N} }$ to be the following:

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

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 each digit of the number. A natural number ${\displaystyle n}$ is a ${\displaystyle b}$-factorion if it is a fixed point for ${\displaystyle SFD_{b}}$, which occurs if ${\displaystyle SFD_{b}(n)=n}$.[7] ${\displaystyle 1}$ and ${\displaystyle 2}$ are fixed points for all ${\displaystyle b}$, 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 factorial of the digits is simply the number of digits ${\displaystyle k}$ in the base 2 representation.

A natural number ${\displaystyle n}$ is a sociable factorion if it is a periodic point for ${\displaystyle SFD_{b}}$, where ${\displaystyle SFD_{b}^{k}(n)=n}$ 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 ${\displaystyle SFD_{b}}$, 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 ${\displaystyle n>SFD_{b}(n)}$ until ${\displaystyle n<b^{b}}$. There are a finite number of 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 ${\displaystyle SFD_{b}^{i}(n)}$ to reach a fixed point is the ${\displaystyle SFD_{b}}$ function's persistence of ${\displaystyle n}$, and undefined if it never reaches a fixed point.

The example below implements the sum of the factorial of the digits described in the definition above to search for factorions and cycles in Python.

def factorial(x: int) -> int:
    total = 1
    for i in range(0, x):
        total = total * (i + 1)
    return total

def sfd(x: int, b: int) -> int:
    """Sum of the factorial of the digits."""
    total = 0
    while x > 0:
        total = total + factorial(x % b)
        x = x // b
    return total

def sfd_cycle(x: int, b: int) -> List[int]:
    seen = []
    while x not in seen:
        seen.append(x)
        x = sfd(x, b)
    cycle = []
    while x not in cycle:
        cycle.append(x)
        x = sfd(x, b)
    return cycle

• 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

• Factorion at Wolfram MathWorld