factorion

Let $n$ be a natural number. For a base $b\; >\; 1$, we define the sum of the factorials of the digits{{Citation|last=Gupta|first=Shyam S.|title=Sum of the Factorials of the Digits of Integers|journal=The Mathematical Gazette|publisher=The Mathematical Association|volume=88|issue=512|year=2004|pages=258–261 |jstor=3620841|doi=10.1017/S0025557200174996|s2cid=125854033|doi-access=free}}{{Citation|last=Sloane|first=Neil|website=On-Line Encyclopedia of Integer Sequences|title=A061602|url=https://oeis.org/A061602}} of $n$, $\backslash operatorname\{SFD\}\_b\; :\; \backslash mathbb\{N\}\; \backslash rightarrow\; \backslash mathbb\{N\}$, to be the following:

:$\backslash operatorname\{SFD\}\_b(n)\; =\; \backslash sum\_\{i=0\}^\{k\; -\; 1\}\; d\_i!.$

where $k\; =\; \backslash lfloor\; \backslash log\_b\; n\; \backslash rfloor\; +\; 1$ is the number of digits in the number in base $b$, $n!$ is the factorial of $n$ and

:$d\_i\; =\; \backslash frac\{n\; \backslash bmod\{b^\{i+1\}\}\; -\; n\; \backslash bmod\{b^\{i\}\}\}\{b^\{i\}\}$

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 $\backslash operatorname\{SFD\}\_b$, i.e. if $\backslash operatorname\{SFD\}\_b(n)\; =\; n$.{{Citation|first=Steve|last=Abbott|title=SFD Chains and Factorion Cycles|journal=The Mathematical Gazette|publisher=The Mathematical Association|volume=88|issue=512|year=2004|pages=261–263 |jstor=3620842|doi=10.1017/S002555720017500X|s2cid=99976100}} $1$ 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 $\backslash operatorname\{SFD\}\_b$, where $\backslash operatorname\{SFD\}\_b^k(n)\; =\; n$ 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 $k\; =\; 2$.{{Citation|last=Sloane|first=Neil|website=On-Line Encyclopedia of Integer Sequences|title=A214285|url=https://oeis.org/A214285}}{{Citation|last=Sloane|first=Neil|website=On-Line Encyclopedia of Integer Sequences|title=A254499|url=https://oeis.org/A254499}}

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

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

• $n\_1\; =\; kb\; +\; 1$ is a factorion for $\backslash operatorname\{SFD\}\_b$ for all $k.$

{{Math proof|title=Proof|drop=hidden|proof=

Let the digits of $n\_1\; =\; d\_1\; b\; +\; d\_0$ be $d\_1\; =\; k$, and $d\_0\; =\; 1.$ Then

: $\backslash operatorname\{SFD\}\_b(n\_1)\; =\; d\_1!\; +\; d\_0!$

:: $=\; k!\; +\; 1!$

:: $=\; k(k\; -\; 1)!\; +\; 1$

:: $=\; d\_1\; b\; +\; d\_0$

:: $=\; n\_1$

Thus $n\_1$ is a factorion for $F\_b$ for all $k$.

}}

• $n\_2\; =\; kb\; +\; 2$ is a factorion for $\backslash operatorname\{SFD\}\_b$ for all $k$.

{{Math proof|title=Proof|drop=hidden|proof=

Let the digits of $n\_2\; =\; d\_1\; b\; +\; d\_0$ be $d\_1\; =\; k$, and $d\_0\; =\; 2$. Then

: $\backslash operatorname\{SFD\}\_b(n\_2)\; =\; d\_1!\; +\; d\_0!$

:: $=\; k!\; +\; 2!$

:: $=\; k(k\; -\; 1)!\; +\; 2$

:: $=\; d\_1\; b\; +\; d\_0$

:: $=\; n\_2$

Thus $n\_2$ is a factorion for $F\_b$ for all $k$.

}}

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

• $n\_1\; =\; b\; +\; k$ is a factorion for $\backslash operatorname\{SFD\}\_b$ for all $k$.

{{Math proof|title=Proof|drop=hidden|proof=

Let the digits of $n\_1\; =\; d\_1\; b\; +\; d\_0$ be $d\_1\; =\; 1$, and $d\_0\; =\; k$. Then

: $\backslash operatorname\{SFD\}\_b(n\_1)\; =\; d\_1!\; +\; d\_0!$

:: $=\; 1!\; +\; k!$

:: $=\; k!\; +\; 1\; -\; k\; +\; k$

:: $=\; 1(k!\; -\; k\; +\; 1)\; +\; k$

:: $=\; d\_1\; b\; +\; d\_0$

:: $=\; n\_1$

Thus $n\_1$ is a factorion for $F\_b$ for all $k$.

}}

All numbers are represented in base $b$.

• 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

{{reflist}}

• [http://mathworld.wolfram.com/Factorion.html Factorion at Wolfram MathWorld]

{{Classes of natural numbers}}

Category:Arithmetic dynamics

Category:Base-dependent integer sequences