Superfactorial Knowpia

In mathematics, and more specifically number theory, the superfactorial of a positive integer $n$ is the product of the first $n$ factorials. They are a special case of the Jordan–Pólya numbers, which are products of arbitrary collections of factorials.

Definition edit

The $n$ th superfactorial ${\mathit {sf}}(n)$ may be defined as:^[1]

{\begin{aligned}{\mathit {sf}}(n)&=1!\cdot 2!\cdot \cdots n!=\prod _{i=1}^{n}i!=n!\cdot {\mathit {sf}}(n-1)\\&=1^{n}\cdot 2^{n-1}\cdot \cdots n=\prod _{i=1}^{n}i^{n+1-i}.\\\end{aligned}}

Following the usual convention for the empty product, the superfactorial of 0 is 1. The sequence of superfactorials, beginning with

{\mathit {sf}}(0)=1

, is:^[1]

1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, ... (sequence A000178 in the OEIS)

Properties edit

Just as the factorials can be continuously interpolated by the gamma function, the superfactorials can be continuously interpolated by the Barnes G-function.^[2]

According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when $p$ is an odd prime number

{\mathit {sf}}(p-1)\equiv (p-1)!!{\pmod {p}},

where

!!

is the notation for the double factorial.^[3]

For every integer $k$ , the number ${\mathit {sf}}(4k)/(2k)!$ is a square number. This may be expressed as stating that, in the formula for ${\mathit {sf}}(4k)$ as a product of factorials, omitting one of the factorials (the middle one, $(2k)!$ ) results in a square product.^[4] Additionally, if any $n+1$ integers are given, the product of their pairwise differences is always a multiple of ${\mathit {sf}}(n)$ , and equals the superfactorial when the given numbers are consecutive.^[1]

References edit

^ ^a ^b ^c Sloane, N. J. A. (ed.), "Sequence A000178 (Superfactorials: product of first n factorials)", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
^ Barnes, E. W. (1900), "The theory of the $G$ -function", The Quarterly Journal of Pure and Applied Mathematics, 31: 264–314, JFM 30.0389.02
^ Aebi, Christian; Cairns, Grant (2015), "Generalizations of Wilson's theorem for double-, hyper-, sub- and superfactorials", The American Mathematical Monthly, 122 (5): 433–443, doi:10.4169/amer.math.monthly.122.5.433, JSTOR 10.4169/amer.math.monthly.122.5.433, MR 3352802, S2CID 207521192
^ White, D.; Anderson, M. (October 2020), "Using a superfactorial problem to provide extended problem-solving experiences", PRIMUS, 31 (10): 1038–1051, doi:10.1080/10511970.2020.1809039, S2CID 225372700

External links edit

Weisstein, Eric W., "Superfactorial", MathWorld

[oeis-1] Sloane, N. J. A. (ed.), "Sequence A000178 (Superfactorials: product of first n factorials)", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation

[barnes-2] Barnes, E. W. (1900), "The theory of the $G$ -function", The Quarterly Journal of Pure and Applied Mathematics, 31: 264–314, JFM 30.0389.02

[wilson-3] Aebi, Christian; Cairns, Grant (2015), "Generalizations of Wilson's theorem for double-, hyper-, sub- and superfactorials", The American Mathematical Monthly, 122 (5): 433–443, doi:10.4169/amer.math.monthly.122.5.433, JSTOR 10.4169/amer.math.monthly.122.5.433, MR 3352802, S2CID 207521192

[square-4] White, D.; Anderson, M. (October 2020), "Using a superfactorial problem to provide extended problem-solving experiences", PRIMUS, 31 (10): 1038–1051, doi:10.1080/10511970.2020.1809039, S2CID 225372700

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Summary

Definition edit

Properties edit

References edit

External links edit