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Superfactorial

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Product of consecutive factorial numbers

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

Definition

The n {\displaystyle n} th superfactorial s f ( n ) {\displaystyle {\mathit {sf}}(n)} may be defined as: s f ( n ) = 1 ! ⋅ 2 ! ⋅ ⋯ n ! = ∏ i = 1 n i ! = n ! ⋅ s f ( n − 1 ) = 1 n ⋅ 2 n − 1 ⋅ ⋯ n = ∏ i = 1 n i n + 1 − i . {\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 s f ( 0 ) = 1 {\displaystyle {\mathit {sf}}(0)=1} , is:

1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, ... (sequence A000178 in the OEIS)

Properties

Just as the factorials can be continuously interpolated by the gamma function, the superfactorials can be continuously interpolated by the Barnes G-function.

According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when p {\displaystyle p} is an odd prime number s f ( p − 1 ) ≡ ( p − 1 ) ! ! ( mod p ) , {\displaystyle {\mathit {sf}}(p-1)\equiv (p-1)!!{\pmod {p}},} where ! ! {\displaystyle !!} is the notation for the double factorial.

For every integer k {\displaystyle k} , the number s f ( 4 k ) / ( 2 k ) ! {\displaystyle {\mathit {sf}}(4k)/(2k)!} is a square number. This may be expressed as stating that, in the formula for s f ( 4 k ) {\displaystyle {\mathit {sf}}(4k)} as a product of factorials, omitting one of the factorials (the middle one, ( 2 k ) ! {\displaystyle (2k)!} ) results in a square product. Additionally, if any n + 1 {\displaystyle n+1} integers are given, the product of their pairwise differences is always a multiple of s f ( n ) {\displaystyle {\mathit {sf}}(n)} , and equals the superfactorial when the given numbers are consecutive.

References

  1. ^ Sloane, N. J. A. (ed.), "Sequence A000178 (Superfactorials: product of first n factorials)", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
  2. 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
  3. 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
  4. 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

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