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Superfactorial

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The superfactorial of a positive integer n is defined by Pickover (1995) as

n$=n!^(n!^(·^(·^(·^(n!)))))_()_(n!).
(1)

The first two values are 1 and 4, but subsequently grow so rapidly that 3$ already has a huge number of digits.

Superfactorial

For a nonnegative integer n, Sloane and Plouffe (1995) define the superfactorial by

n$=product_(k=1)^(n)k!
(2)
=G(n+2),
(3)

which is equivalent to the integral values of the Barnes G-function. The values for n=0, 1, ... are 1, 1, 2, 12, 288, 34560, ... (OEIS A000178). This function has an unexpected connection with Bell numbers.

See also

Barnes G-Function, Bell Number, Factorial, Hyperfactorial, Large Number, Subfactorial, Vandermonde Determinant

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References

Fletcher, A.; Miller, J. C. P.; Rosenhead, L.; and Comrie, L. J. An Index of Mathematical Tables, Vol. 1. Oxford, England: Blackwell, p. 50, 1962.Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, p. 231, 1994.Numberphile. "What's Special About 288?" Oct. 24, 2023. https://www.youtube.com/watch?v=xV4A8oU3yew.Pickover, C. A. Keys to Infinity. New York: Wiley, p. 102, 1995.Radoux, C. "Query 145." Not. Amer. Math. Soc. 25, 197, 1978.Ryser, H. J. Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., p. 53, 1963.Sloane, N. J. A. Sequence A000178/M2049 in "The On-Line Encyclopedia of Integer Sequences."Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.

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Superfactorial

Cite this as:

Weisstein, Eric W. "Superfactorial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Superfactorial.html

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