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 $n$, $\operatorname_b : \mathbb \rightarrow \mathbb$, to be the following:
:$\operatorname_b(n) = \sum_^ d_i!.$
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
:$d_i = \frac$
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 $\operatorname_b$, i.e. if $\operatorname_b(n) = n$. $1$ 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 $\operatorname_b$, where $\operatorname_b^k(n) = n$ for a positive integer $k$, and forms a cycle of period $k$. A factorion is a sociable factorion with $k = 1$, and a amicable factorion is a sociable factorion with $k = 2$.

All natural numbers $n$ are preperiodic points for $\operatorname_b$, regardless of the base. This is because all natural numbers of base $b$ with $k$ digits satisfy $b^ \leq n \leq (b-1)!(k)$. However, when $k \geq b$, then $b^ > (b-1)!(k)$ for $b > 2$, so any $n$ will satisfy $n > \operatorname_b(n)$ 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 $\operatorname_b^i(n)$ to reach a fixed point is the $\operatorname_b$ function's persistence of $n$, and undefined if it never reaches a fixed point.

Factorions for

''b'' = (''k'' − 1)!

Let $k$ be a positive integer and the number base $b = (k - 1)!$. Then:
*$n_1 = kb + 1$ is a factorion for $\operatorname_b$ for all $k.$

*$n_2 = kb + 2$ is a factorion for $\operatorname_b$ for all $k$.

''b'' = ''k''! − ''k'' + 1

Let $k$ be a positive integer and the number base $b = k! - k + 1$. Then:
*$n_1 = b + k$ is a factorion for $\operatorname_b$ for all $k$.

Table of factorions and cycles of

All numbers are represented in base $b$.

Programming example

The example below implements the sum of the factorials 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 factorials 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 nt
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

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

References

External links

Factorion at Wolfram MathWorld

{{Classes of natural numbers

Arithmetic dynamics
Base-dependent integer sequences