Superfactorial
In mathematics, and more specifically number theory, the superfactorial of a positive integer [math]\displaystyle{ n }[/math] is the product of the first [math]\displaystyle{ n }[/math] factorials. They are a special case of the Jordan–Pólya numbers, which are products of arbitrary collections of factorials.
Definition
The [math]\displaystyle{ n }[/math]th superfactorial [math]\displaystyle{ \mathit{sf}(n) }[/math] may be defined as:[1] [math]\displaystyle{ \begin{align} \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{align} }[/math] Following the usual convention for the empty product, the superfactorial of 0 is 1. The sequence of superfactorials, beginning with [math]\displaystyle{ \mathit{sf}(0)=1 }[/math], is:[1]
Properties
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 [math]\displaystyle{ p }[/math] is an odd prime number [math]\displaystyle{ \mathit{sf}(p-1)\equiv(p-1)!!\pmod{p}, }[/math] where [math]\displaystyle{ !! }[/math] is the notation for the double factorial.[3]
For every integer [math]\displaystyle{ k }[/math], the number [math]\displaystyle{ \mathit{sf}(4k)/(2k)! }[/math] is a square number. This may be expressed as stating that, in the formula for [math]\displaystyle{ \mathit{sf}(4k) }[/math] as a product of factorials, omitting one of the factorials (the middle one, [math]\displaystyle{ (2k)! }[/math]) results in a square product.[4] Additionally, if any [math]\displaystyle{ n+1 }[/math] integers are given, the product of their pairwise differences is always a multiple of [math]\displaystyle{ \mathit{sf}(n) }[/math], and equals the superfactorial when the given numbers are consecutive.[1]
References
- ↑ 1.0 1.1 1.2 Sloane, N. J. A., ed. "Sequence A000178 (Superfactorials: product of first n factorials)". OEIS Foundation. https://oeis.org/A000178.
- ↑ [268,%22view%22:%22toc%22} "The theory of the G-function"], The Quarterly Journal of Pure and Applied Mathematics 31: 264–314, 1900, https://gdz.sub.uni-goettingen.de/id/PPN600494829_0031?tify={%22pages%22:[268],%22view%22:%22toc%22}
- ↑ 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
- ↑ 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
External links
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