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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]

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

1. SpongeTechX, Googology World
2. Factorexation - Tech's Large Numbers