Definition:Superfactorial

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Definition

Let $n \in \Z_{\ge 0}$ be a positive integer.


The superfactorial of $n$ is defined as:

$n\$ = \displaystyle \prod_{k \mathop = 1}^n k! = 1! \times 2! \times \cdots \times \left({n - 1}\right)! \times n!$

where $k!$ denotes the factorial of $n$.


Sequence

The sequence of superfactorials begins:

$1, 2, 12, 288, 34 \, 560, 24 \, 883 \, 200, 125 \, 411 \, 328 \, 000, \ldots$

This sequence is A000178 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Also defined as

Some sources, for example Clifford A. Pickover, define the superfactorial as:

$n \$ = \underbrace{n!^{n!^{·^{·^{.^{n!}}}}} }_n$


Examples

Superfactorial of $1$

The superfactorial of $1$ is given by:

$1 \$ = 1! = 1$


Superfactorial of $2$

The superfactorial of $2$ is given by:

$2 \$ = 2!^{2!} = 2^2 = 4$


Superfactorial of $3$

The superfactorial of $3$ is given by:

$3 \$ = 3!^{3!^{3!} } = 6^{6^6} = 6^{46 \, 656}$


Also see

  • Results about Superfactorials can be found here.


Sources