Superfactorial

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

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

The above it is also equal to  or in up-arrow notation. The first few values of  Clifford Pickover superfactorial  for \(n = 1, 2, 3, 4, \ldots\)   are :

\(1$ = 1\)

\(2$ = 4\)

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

\(4$ = 22pt33.265015361457.. = 23pt1.52..\)

\(4$ = 22pt33. 265015361457..  = 23pt 1.521987728335 ..\)

N.J.A. Sloane and Simon Plouffe define it 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, ....

The latter 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}