The term **superfactorial **has two slightly different definitions: as a product of factorials (Sloane & Plouffe, 1995) or as a tower of factorials involving compound exponents (Pickover, 1995). Sloan and Pouffe’s form is the most common.

## 1. Sloane & Plouffe’s Superfactorial

A **superfactorial **is defined by Sloane and Plouffe as the product of the first *n *factorials. Their definition is equivalent to the integral values for the Barnes G-function.

## Examples

The superfactorial of 3 is:

sf(3) = 1 factorial * 2 factorial * 3 factorial = 1 * 2 * 6 = 12

Superfactorials for integers 1 through 11 are (OEIS A000178):

- 1
- 2
- 12
- 288
- 34560
- 24883200
- 125411328000
- 5056584744960000
- 1834933472251084800000
- 6658606584104736522240000000
- 265790267296391946810949632000000000
- 127313963299399416749559771247411200000000000

## Pickover’s Superfactorial

Pickover (1995) defines a different superfactorial, one that involves compound exponentiation:

The dollar sign ($) is actually a factorial symbol (an exclamation mark !) overwritten with the letter S (Mudunuru et al., 2017).

This can also be expressed as a tetration:

n$ = ^{n!} (n!).

Alternatively, it can be expressed as a tower of exponents, using **Knuth’s arrow up notation**:

So *a* ↑↑*n*, is iterated exponentiation (i.e.tetration), and means to raise *a* to itself *n* – 1 times. For example,

a ↑↑5 = a^{aaaa}.

## Examples

The first two values are:

- 1$ = 1,
- 2$ = 3

From n = 3 this grows very rapidly and up the numbers are *huge*. $n is roughly 10^{101036305}.

## 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.

Mudunuru et al. Zero Factorial. Sch. J. Phys. Math. Stat. 2017; 4(4):172-177

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

Ryser, H. J. Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., p. 53, 1963.

Sloane, Plouffe. The Encyclopedia of Integer Sequences. Academic Press, 1995

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