Superfactorial

The superfactorial of $n$ is defined by Pickover (1995) as

$$n\$\equiv \underbrace{{n!}^{{n!}^{\cdot^{\cdot^{\cdot^{n!}}}}}\!\!}_{n!}.$$

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

Sloane and Plouffe (1995) define the superfactorial by

$$n\$\equiv \prod_{i=1}^n i!,$$

which is equivalent to the integral values of the G-Function. The first few values are 1, 1, 2, 12, 288, 34560, ... (Sloane's A000178).

See also Factorial, G-Function, Large Number, Subfactorial

References

Pickover, C. A. Keys to Infinity. New York: Wiley, p. 102, 1995.

Sloane, N. J. A. Sequence A000178/M2049 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.