Superfactorial
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]
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
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
- ^ 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
- JFM 30.0389.02
- S2CID 207521192
- S2CID 225372700