Superfactorial

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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.

Contents

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]

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. [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]

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References

  1. 1 2 3 Sloane, N. J. A. (ed.), "SequenceA000178(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