In mathematics, and more specifically number theory, the superfactorial of a positive integer is the product of the first factorials. They are a special case of the Jordan–Pólya numbers, which are products of arbitrary collections of factorials.
Definition
The th superfactorial may be defined as:[1]
Following the usual convention for the empty product, the superfactorial of 0 is 1. The sequence of superfactorials, beginning with , 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 is an odd prime number
where is the notation for the double factorial.[3]
For every integer , the number is a square number. This may be expressed as stating that, in the formula for as a product of factorials, omitting one of the factorials (the middle one, ) results in a square product.[4] Additionally, if any integers are given, the product of their pairwise differences is always a multiple of , and equals the superfactorial when the given numbers are consecutive.[1]
References
- 1 2 3 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