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
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):
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 = aaaaa.
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 10101036305.
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
Stephanie Glen. "Superfactorial: Definition (Sloane, Pickover’s)" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/superfactorial-definition-sloane-pickovers/
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