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

Definition

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

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

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

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 ${\displaystyle p}$ is an odd prime number

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

${\displaystyle !!}$

For every integer ${\displaystyle k}$, the number ${\displaystyle {\mathit {sf))(4k)/(2k)!}$ 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, ${\displaystyle (2k)!}$) results in a square product.^[4] Additionally, if any ${\displaystyle n+1}$ integers are given, the product of their pairwise differences is always a multiple of ${\displaystyle {\mathit {sf))(n)}$, and equals the superfactorial when the given numbers are consecutive.^[1]