Factorion

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]

Definition

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.

Factorions for ${\displaystyle SFD_{b}}$

b = (k - 1)!

Let ${\displaystyle k}$ be a positive integer and the number base ${\displaystyle b=(k-1)!}$. Then:

• ${\displaystyle n_{1}=kb+1}$ is a factorion for ${\displaystyle SFD_{b}}$ for all ${\displaystyle k}$.

Proof —

Let the digits of ${\displaystyle n_{1}=d_{1}b+d_{0}}$ be ${\displaystyle d_{1}=k}$, and ${\displaystyle d_{0}=1}$. Then

${\displaystyle SFD_{b}(n_{1})=d_{1}!+d_{0}!}$

${\displaystyle =k!+1!}$

${\displaystyle =k(k-1)!+1}$

${\displaystyle =d_{1}b+d_{0}}$

${\displaystyle =n_{1}}$

Thus ${\displaystyle n_{1}}$ is a factorion for ${\displaystyle F_{b}}$ for all ${\displaystyle k}$.

• ${\displaystyle n_{2}=kb+2}$ is a factorion for ${\displaystyle SFD_{b}}$ for all ${\displaystyle k}$.

Proof —

Let the digits of ${\displaystyle n_{2}=d_{1}b+d_{0}}$ be ${\displaystyle d_{1}=k}$, and ${\displaystyle d_{0}=2}$. Then

${\displaystyle SFD_{b}(n_{2})=d_{1}!+d_{0}!}$

${\displaystyle =k!+2!}$

${\displaystyle =k(k-1)!+2}$

${\displaystyle =d_{1}b+d_{0}}$

${\displaystyle =n_{2}}$

Thus ${\displaystyle n_{2}}$ is a factorion for ${\displaystyle F_{b}}$ for all ${\displaystyle k}$.

Factorions

${\displaystyle k}$	${\displaystyle b}$	${\displaystyle n_{1}}$	${\displaystyle n_{2}}$
4	6	41	42
5	24	51	52
6	120	61	62
7	720	71	72

b = k! - k + 1

Let ${\displaystyle k}$ be a positive integer and the number base ${\displaystyle b=k!-k+1}$. Then:

• ${\displaystyle n_{1}=b+k}$ is a factorion for ${\displaystyle SFD_{b}}$ for all ${\displaystyle k}$.

Proof —

Let the digits of ${\displaystyle n_{1}=d_{1}b+d_{0}}$ be ${\displaystyle d_{1}=1}$, and ${\displaystyle d_{0}=k}$. Then

${\displaystyle SFD_{b}(n_{1})=d_{1}!+d_{0}!}$

${\displaystyle =1!+k!}$

${\displaystyle =k!+1-k+k}$

${\displaystyle =1(k!-k+1)+k}$

${\displaystyle =d_{1}b+d_{0}}$

${\displaystyle =n_{1}}$

Thus ${\displaystyle n_{1}}$ is a factorion for ${\displaystyle F_{b}}$ for all ${\displaystyle k}$.

Factorions

${\displaystyle k}$	${\displaystyle b}$	${\displaystyle n_{1}}$
3	4	13
4	21	14
5	116	15
6	715	16

Table of factorions and cycles of ${\displaystyle SFD_{b}}$

All numbers are represented in base ${\displaystyle b}$.

Base ${\displaystyle b}$	Nontrivial factorion (${\displaystyle n\neq 1}$, ${\displaystyle n\neq 2}$)[10]	Cycles
2	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$
3	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$
4	13	3 → 12 → 3
5	144	${\displaystyle \varnothing }$
6	41, 42	${\displaystyle \varnothing }$
7	${\displaystyle \varnothing }$	36 → 2055 → 465 → 2343 → 53 → 240 → 36
8	${\displaystyle \varnothing }$	3 → 6 → 1320 → 12 175 → 12051 → 175
9	62558
10	145, 40585	871 → 45361 → 871[9] 872 → 45362 → 872[8]

Programming example

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

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

1. Sloane, Neil, "A014080", On-Line Encyclopedia of Integer Sequences
2. Gardner, Martin (1978), "Factorial Oddities", Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-Of-Mind, Vintage Books, pp. 61 and 64, ISBN 9780394726236
3. Madachy, Joseph S. (1979), Madachy's Mathematical Recreations, Dover Publications, p. 167, ISBN 9780486237626
4. Pickover, Clifford A. (1995), "The Loneliness of the Factorions", Keys to Infinity, John Wiley & Sons, pp. 169–171 and 319–320, ISBN 9780471193340 – via Google Books
5. Gupta, Shyam S. (2004), "Sum of the Factorials of the Digits of Integers", The Mathematical Gazette, The Mathematical Association, 88 (512): 258–261, doi:10.1017/S0025557200174996, JSTOR 3620841
6. Sloane, Neil, "A061602", On-Line Encyclopedia of Integer Sequences
7. Abbott, Steve (2004), "SFD Chains and Factorion Cycles", The Mathematical Gazette, The Mathematical Association, 88 (512): 261–263, doi:10.1017/S002555720017500X, JSTOR 3620842
8. Sloane, Neil, "A214285", On-Line Encyclopedia of Integer Sequences
9. Sloane, Neil, "A254499", On-Line Encyclopedia of Integer Sequences
10. Sloane, Neil, "A193163", On-Line Encyclopedia of Integer Sequences

External links

• Factorion at Wolfram MathWorld