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. 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.

Factorions for 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 ${\displaystyle \operatorname {SFD} _{b}}$ 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

${\displaystyle \operatorname {SFD} _{b}(n_{1})=d_{1}!+d_{0}!}$

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

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

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

${\displaystyle =n_{1}}$

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

• ${\displaystyle n_{2}=kb+2}$ is a factorion for ${\displaystyle \operatorname {SFD} _{b}}$ 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

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

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

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

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

${\displaystyle =n_{2}}$

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 ${\displaystyle \operatorname {SFD} _{b}}$ 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

${\displaystyle \operatorname {SFD} _{b}(n_{1})=d_{1}!+d_{0}!}$

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

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

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

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

${\displaystyle =n_{1}}$

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 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]

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, S2CID   125854033
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, S2CID   99976100
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