Factorexation

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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 "TechKon" (formerly SpongeTechX).^[1]^[2] or mostly know as "fz-n-factorial"

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 notations[]

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

Examples[]

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!)!} ~ 10^{1.012*10^197,582}
(4 \)\ = (4!^4!)!^{(4!^4!)!} ~ 10^{10^4.36*10³⁴}