Definition:Superfactorial
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Definition
Let $n \in \Z_{\ge 0}$ be a positive integer.
The superfactorial of $n$ is defined as:
- $n\$ = \ds \prod_{k \mathop = 1}^n k! = 1! \times 2! \times \cdots \times \left({n - 1}\right)! \times n!$
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