Definition:Superfactorial

Definition

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

The superfactorial of $n$ is defined as:

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

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

Sequence

The sequence of superfactorials begins:

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

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

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $288$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $3!^{3!^{3!} }$

• Weisstein, Eric W. "Superfactorial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Superfactorial.html