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Factorexation refers to the operation \(n\,\backslash\) (pronounced "\(n\) factorexated"), defined as

\[n\,\backslash = n!^{n!} = n! \uparrow\uparrow 2.\]

The term was coined by a Googology Wiki user under the alias of "SpongeTechX".

Iterated factorexation is written with multiple backslashes, e.g. \(n\,\backslash\backslash\backslash\). It can also be abbreviated \(n\,\backslash^k\) when there are \(k\) iterations. This function can be recursively defined as follows: \(n\backslash^0=n\) and \(n\backslash^{k+1}=(n\backslash^k)!^{(n\backslash^k)!}\)

Approximations in other notationsEdit

Notation Approximation
Fast-growing hierarchy \(f_2(f_2(n))\)
Hardy hierarchy \(H_{\omega^2 2}(n)\)
Slow-growing hierarchy \(g_{\omega^{\omega^\omega}}(n)\)

ExamplesEdit

The following are examples of factorexating a number once:

  • 2 \ = 2!2! = 4
  • 3 \ = 3!3! = 66 = 46,656
  • 4 \ = 4!4! = 2424 = 1,333,735,776,850,284,124,449,081,472,843,776
  • 5 \ = 5!5! = 120120 = 3.1750423737803369*10249

The following are examples of factorexating a number twice:

  • 2 \\ = (2!2!)!(2!2!)! = 2424 = 1,333,735,776,850,284,124,449,081,472,843,776 (same as 4 \)
  • 3 \\ = (3!3!)!(3!3!)! ~ 101.012*10197,582
  • 4 \\ = (4!4!)!(4!4!)! ~ 10104.36*1034

SourcesEdit

See also Edit

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 · Bouncing Factorial
Tetrational growth: Exponential factorial · Expostfacto function · Superfactorial by Clifford Pickover
Array-based extensions: Hyperfactorial array notation · Nested factorial notation
Other googological variants: · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial · Bouncing Factorial

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