Factorexation

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 "" (formerly SpongeTechX).SpongeTechX, Googology WorldFactorexation - Tech's Large Numbers 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!)! ~ 101.012*10197,582
* (4 \)\ = (4!4!)!(4!4!)! ~ 10104.36*1034
==Sources==
== See also ==
Category:Functions
Category:Factorials
Category:SpongeTechX