In January 25th 2016 I was editing the article about superfactorial, I made a few corrections and added some values for \(n\$\) as defined by Clifford Pickover.

I was reading about the two different definitions of superfactorial in the article, then I realized what would be a third definition for \(n\$\), so I decided to create a new factorial-based function.

My inttention was to create a nice new \(n\$\) function that grew faster than that defined by Sloane & Plouffeand that does not involve tetration, as observed in \(n\$\) reported by Clifford Pickover.

Considering the product of factorials as in \(n\$\) reported by Sloane & Plouffe, after some attempts, I got through a new definition for \(n\$\) as follows:

\(n\$ = (\underbrace{11...11}_{n}n)\times((\underbrace{11...11}_{n-1}n)!)\times((\underbrace{11...11}_{n-2}n)!^{2})\cdots((111n)!^{(n-3)})\times((11n)!^{(n-2)})\times(n!^{(n-1)})\)

where \(!^{2}\), \(!^{(n-3)}\), \(!^{(n-2)}\) and \(!^{(n-1)}\) are from Nested factorial notation as defined by Aarex Tiaokhiao.

The first six superfactorials as described here are:

\(1\$ = 1\)

\(2\$ = 44\)

\(3\$ = 2.08191223228633...\times10^{42} = 2pt1.62652987982134...\)

\(4\$ = 10^{10^{56.15691512713508...}} = 2pt56.15691512713508... = 3pt1.74940324263919...\)

\(5\$ = 4pt2.44662641649391...\)

\(6\$ = 5pt3.24295297233703...\)