Hyperfactorials

Occasionally the hyperfactorial of n is considered. It is written as H(n) and defined by

For n = 1, 2, 3, 4, ... the values H(n) are 1, 4, 108, 27648,... (sequence A002109 in OEIS).

The asymptotic growth rate is

where A = 1.2824... is the Glaisher–Kinkelin constant.[9] H(14) = 1.8474...×1099 is already almost equal to a googol, and H(15) = 8.0896...×10116 is almost of the same magnitude as the Shannon number, the theoretical number of possible chess games. Compared to the Pickover definition of the superfactorial, the hyperfactorial grows relatively slowly. The hyperfactorial function can be generalized to complex numbers in a similar way as the factorial function. The resulting function is called the K-function.