The weak factorial is a factorial-related function so-named by Cookie Fonster. It is equal to the least smallest number divisible by all numbers 1 through x. [1]


\(wf(x) = LCM(x, wf(x-1))\)

\(wf(1) = 1\)

The first ten weak factorial numbers are 1, 2, 6, 12, 60, 60, 420, 840, 2,520, and 2,520.

It can be shown that value of this function increases only at arguments which are prime powers. Because of that, there will be long runs where the function is constant.

This function can be shown to be equal to \(e^{\psi(x)}\), where \(\psi(x)\) is the second Chebyshev function, so as a collorary from prime number theorem, it can be approximated by ex.

Specific numbers

  • 420 is an integer equal to the weak factorial of 7, and also the number of possible king moves in chess.[citation needed]
  • 840 is an integer equal to the weak factorial of 8, and also the largest known kissing number in 12 dimensions.[citation needed]
  • 2520 is an integer equal to the weak factorial of 10, and is part of the 53-aliquot tree. The complete aliquot sequence starting at 1080 is: 1080, 2520, 6840, 16560, 41472, 82311, 27441, 12209, 451, 53, 1, 0.[citation needed]


External links

See also

Main article: Factorial
Multifactorials: Double factorial · Multifactorial
Falling and rising: Falling factorial · Rising factorial
Other mathematical variants: Alternating factorial · Hyperfactorial · q-factorial · Roman factorial · Subfactorial · Weak factorial · Primorial · Compositorial · Semiprimorial
Tetrational growth: Exponential factorial · Expostfacto function · Superfactorial by Clifford Pickover
Nested Factorials: Tetorial · Petorial · Ectorial · Zettorial · Yottorial
Array-based extensions: Hyperfactorial array notation · Nested factorial notation
Other googological variants: · Tetrofactorial · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial · Bouncing Factorial
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