Supremefactorial

The supremefactorial is a factorial-based function. This function was defined by Twikie.Twikie, Supremefactorial! Accessed 2023-05-15.
== Definition ==
Supremefactorial, denoted by \(n§\), is defined as follows:
\(n§=\)\((^n(^nn))!\)
The following is the explanation of the use of supremefactorial:
Firstly, you count the partial result that's in the inner parenthesis, for example, let's say that \(n = 3\), this makes \(^nn =\) \(^33\). That's basically \(3^{3^{3}}\) which equals to \(7,625,597,484,987\).
Secondly, you count the partial result that's in the outer parenthesis. Since we're using \(n = 3\), and the first partial result was \(7,625,597,484,987\), you need to put \(7,625,597,484,987\) onto its third hyperpower. After doing it, it should look something like this: \(7,625,597,484,987^{7,625,597,484,987^{7,625,597,484,987}}\).
Thirdly, you factor the main parital result, and it should give you the answer.
If you want approximations, you can follow the steps that are noted on the bottom of the page! 
== Approximations ==
The approximations of the first values of the supremefactorial are:
*\(1§=1\)
*\(2§\approx 8.5781777534284\times 10^{506}\)
*\(3§\approx 10^{10^{10^{10^{13.99226640545933}}}}\)
*\(4§\approx 10^{10^{10^{10^{10^{154.2080275436318}}}}}\)
*\(5§\approx 10^{10^{10^{10^{10^{10^{2184.125722088846}}}}}}\)
*\(6§\approx 10^{10^{10^{10^{10^{10^{10^{36305.31580191888}}}}}}}\)
*\(7§\approx 10^{10^{10^{10^{10^{10^{10^{10^{10^{5.842593328962332}}}}}}}}}\)
\(n§\) can be approximated by up-arrow notation and down-arrow notation with the lower and upper bound \(n\uparrow\uparrow\downarrow 3\) and \((n\uparrow\uparrow\downarrow 3)\uparrow\uparrow 2\).
==Sources==
Category:Factorials
Category:Notations
Category:Videos