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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.^[1]^[2]^[3] The name factorion was coined by the author Clifford A. Pickover.^[4]

Definition

Let $n$ be a natural number. For a base $b>1$ , we define the sum of the factorials of the digits^[5]^[6] of $n$ , $\operatorname {SFD} _{b}:\mathbb {N} \rightarrow \mathbb {N}$ , to be the following:

\operatorname {SFD} _{b}(n)=\sum _{i=0}^{k-1}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 {n{\bmod {b^{i+1}}}-n{\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 $\operatorname {SFD} _{b}$ , i.e. if $\operatorname {SFD} _{b}(n)=n$ .^[7] $1$ 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 $\operatorname {SFD} _{b}$ , where $\operatorname {SFD} _{b}^{k}(n)=n$ 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 $k=2$ .^[8]^[9]

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

Factorions for $SFD b$

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 $\operatorname {SFD} _{b}$ for all $k.$

Proof

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

\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 $\operatorname {SFD} _{b}$ for all $k$ .

Proof

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

\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 $k$ .

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

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 $\operatorname {SFD} _{b}$ for all $k$ .

Proof

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

\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 $k$ .

Factorions
$k$	$b$	$n_{1}$
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 $b$ .

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

References

^ Sloane, Neil, "A014080", On-Line Encyclopedia of Integer Sequences
^ 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
^ Madachy, Joseph S. (1979), Madachy's Mathematical Recreations, Dover Publications, p. 167, ISBN 9780486237626
^ 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
^ Gupta, Shyam S. (2004), "Sum of the Factorials of the Digits of Integers", The Mathematical Gazette, 88 (512), The Mathematical Association: 258–261, doi:10.1017/S0025557200174996, JSTOR 3620841, S2CID 125854033
^ Sloane, Neil, "A061602", On-Line Encyclopedia of Integer Sequences
^ Abbott, Steve (2004), "SFD Chains and Factorion Cycles", The Mathematical Gazette, 88 (512), The Mathematical Association: 261–263, doi:10.1017/S002555720017500X, JSTOR 3620842, S2CID 99976100
^ ^a ^b Sloane, Neil, "A214285", On-Line Encyclopedia of Integer Sequences
^ ^a ^b Sloane, Neil, "A254499", On-Line Encyclopedia of Integer Sequences
^ Sloane, Neil, "A193163", On-Line Encyclopedia of Integer Sequences

External links

Factorion at Wolfram MathWorld

[A014080-1] Sloane, Neil, "A014080", On-Line Encyclopedia of Integer Sequences

[Gardner-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

[Madachy-3] Madachy, Joseph S. (1979), Madachy's Mathematical Recreations, Dover Publications, p. 167, ISBN 9780486237626

[Pickover-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

[Gupta-5] Gupta, Shyam S. (2004), "Sum of the Factorials of the Digits of Integers", The Mathematical Gazette, 88 (512), The Mathematical Association: 258–261, doi:10.1017/S0025557200174996, JSTOR 3620841, S2CID 125854033

[A061602-6] Sloane, Neil, "A061602", On-Line Encyclopedia of Integer Sequences

[Abbott-7] Abbott, Steve (2004), "SFD Chains and Factorion Cycles", The Mathematical Gazette, 88 (512), The Mathematical Association: 261–263, doi:10.1017/S002555720017500X, JSTOR 3620842, S2CID 99976100

[A214285-8] Sloane, Neil, "A214285", On-Line Encyclopedia of Integer Sequences

[A254499-9] Sloane, Neil, "A254499", On-Line Encyclopedia of Integer Sequences

[A193163-10] Sloane, Neil, "A193163", On-Line Encyclopedia of Integer Sequences

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Definition

Factorions for $SFD b$

b = (k − 1)!

b = k! − k + 1

Table of factorions and cycles of $SFD b$

See also

References

External links

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Definition

Factorions for SFDb

b = (k − 1)!

b = k! − k + 1

Table of factorions and cycles of SFDb

See also

References

External links

One thought on “Cannabaceae”

Leave a Reply

Factorions for $SFD b$

Table of factorions and cycles of $SFD b$