Hyperfactorial Last updated May 27, 2025 Definition The hyperfactorial of a positive integer n {\displaystyle n} is the product of the numbers 1 1 , 2 2 , … , n n {\displaystyle 1^{1},2^{2},\dots ,n^{n}} . That is, [ 1] [ 2] H ( n ) = 1 1 ⋅ 2 2 ⋅ ⋯ n n = ∏ i = 1 n i i = n n H ( n − 1 ) . {\displaystyle H(n)=1^{1}\cdot 2^{2}\cdot \cdots n^{n}=\prod _{i=1}^{n}i^{i}=n^{n}H(n-1).} Following the usual convention for the empty product , the hyperfactorial of 0 is 1. The sequence of hyperfactorials, beginning with H ( 0 ) = 1 {\displaystyle H(0)=1} , is: [ 1]
1, 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... (sequence
A002109 in the
OEIS )
Other properties 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 H ( p − 1 ) ≡ ( − 1 ) ( p − 1 ) / 2 ( p − 1 ) ! ! ( mod p ) , {\displaystyle H(p-1)\equiv (-1)^{(p-1)/2}(p-1)!!{\pmod {p}},} where ! ! {\displaystyle !!} is the notation for the double factorial . [ 4]
The hyperfactorials give the sequence of discriminants of Hermite polynomials in their probabilistic formulation. [ 1]
References 1 2 3 Sloane, N. J. A. (ed.), "Sequence A002109 (Hyperfactorials: Product_{k = 1..n} k^k)" , The On-Line Encyclopedia of Integer Sequences , OEIS Foundation 1 2 Alabdulmohsin, Ibrahim M. (2018), Summability Calculus: A Comprehensive Theory of Fractional Finite Sums , Cham: Springer, pp. 5– 6, doi :10.1007/978-3-319-74648-7 , ISBN 978-3-319-74647-0 , MR 3752675 , S2CID 119580816 1 2 Kinkelin, H. (1860), "Ueber eine mit der Gammafunction verwandte Transcendente und deren Anwendung auf die Integralrechung" [ On a transcendental variation of the gamma function and its application to the integral calculus] , Journal für die reine und angewandte Mathematik (in German), 1860 (57): 122– 138, doi :10.1515/crll.1860.57.122 , S2CID 120627417 1 2 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 1 2 Glaisher, J. W. L. (1877), "On the product 11 .22 .33 ... n n " , Messenger of Mathematics , 7 : 43– 47 This page is based on this
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