In mathematics, and more specifically number theory, the superfactorial of a positive integer
They are a special case of the Jordan–Pólya numbers, which are products of arbitrary collections of factorials.
{\displaystyle {\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
{\displaystyle {\mathit {sf}}(0)=1}
, is:[1] 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
{\displaystyle {\mathit {sf}}(p-1)\equiv (p-1)!!
is the notation for the double factorial.
{\displaystyle {\mathit {sf}}(4k)/(2k)!}
is a square number.
This may be expressed as stating that, in the formula for
{\displaystyle {\mathit {sf}}(4k)}
as a product of factorials, omitting one of the factorials (the middle one,
) results in a square product.
integers are given, the product of their pairwise differences is always a multiple of
, and equals the superfactorial when the given numbers are consecutive.