Superfactorial

The superfactorial is a factorial-based function with differing definitions.

Clifford A. Pickover
Clifford Pickover defines superfactorial as \(n\$ = {}^{(n!)}(n!) = \underbrace{n!^{n!^{n!^{.^{.^.}}}}}_{n!}\).

The above it is also equal to  or in up-arrow notation.

Some values of  Clifford Pickover superfactorial  calculated using HyperCalc, WolframAlphaand bcalc are below :

\(1$ = 1\)

\(2$ = 4\)

\(3$ = 10^{10^{10^{10^{36305.315801918918..}}}} = 4pt36305.315801918918.. = 5pt4.559970218821..\)

\(4$ = 24pt1.521987728335..\)

\(5$ = 120pt2.397626581446..\)

\(6$ = 720pt3.313389520154..\)

\(7$ = 5040pt4.270930686287..\)

\(8$ = 40320pt5.268800796659..\)

\(9$ = 362880pt.6.304819474820..\)

\(10$ = 362880pt7.376651198837..\)

\(11$ = 39916800pt8.482035348919..\)

\(12$ = 479001600pt9.618873548666..\)

\(13$ = 6227020801pt1.032830331015..\)

\(14$ = 87178291201pt1.078436584986..\)

\(15$ = 1307674368001pt1.120569877239..\)

\(...\)

\(100$ = (100!+1)pt2.204577320632..\)

\(...\)

 \(1000$ = (1000!+2)pt3.410104470640..\)

 \(...\)

 \(10^{6}$ = (10^{6}!+2)pt6.745521015639..\)

 \(...\)

 \(googol$ = (googol!+2)pt2.008592123510..\)

 \(...\)

 \(...\)

 \(multillion$ = (multillion!+6)p1.628155076210..\) Neil J.A. Sloane & Simon Plouffe

 Sloane and Plouffe define superfactoril as \(n\$ = \prod^{n}_{i = 1} i! = 1! \cdot 2! \cdot 3! \cdot 4! \cdot \ldots \cdot n!\), the product of the first \(n\) factorials. The first few values of \(n$\) for \(n = 1, 2, 3, \ldots\) are 1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, 1834933472251084800000, 6658606584104736522240000000, 265790267296391946810949632000000000, 127313963299399416749559771247411200000000000, ....

This superfactorial has an interesting relationship to the hyperfactorial: \(n\$ \cdot H(n) = n!^{n + 1}\). This may be proven by induction, with the base case \(1\$ \cdot H(1) = 1 = 1!^2\) and the following simple inductive step:

\begin{eqnarray} n\$ \cdot H(n) &=& n!^{n + 1} \\ n\$ \cdot H(n) \cdot (n + 1)! \cdot (n + 1)^{n + 1} &=& n!^{n + 1} \cdot (n + 1)! \cdot (n + 1)^{n + 1} \\ (n + 1)\$ \cdot H(n + 1) &=& (n + 1)!^{n + 2} \\ \end{eqnarray}

Daniel Corrêa
Giving a new definition to superfactorial, Daniel Correacoined the term \(n$\) as below:

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

Sources