Nondivisorial

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The nondivisorial of $\scriptstyle n\,$ is the product of nondivisors of $\scriptstyle n\,$ (product of all positive integers up to $\scriptstyle n\,$ which are not factors of $\scriptstyle n\,$).

By analogy with phi-torial (phitorial) for coprimorial, might be called co-tau-torial of $\scriptstyle n\,$ ($\scriptstyle n\,$ co-tau-torial) or co-tautorial of $\scriptstyle n\,$ ($\scriptstyle n\,$ co-tautorial) since the product involves $\scriptstyle {\overline {\tau }}(n)\,$ numbers, where $\scriptstyle {\overline {\tau }}(n)\,$ is the number of nondivisors of $\scriptstyle n\,$.

Formulae

Thus the nondivisorial of $\scriptstyle n\,$ is

$\Pi _{\overline {\tau }}(n)\equiv \prod _{\stackrel {i=1}{i\nmid n}}^{n}i=\prod _{i=1}^{n}i^{(1-[n{\bmod {i}}\,=\,0])}={\frac {n!}{\Pi _{\tau }(n)}}\,$

where $\scriptstyle {\overline {\tau }}(n)\,$ is the number of nondivisors of $\scriptstyle n\,$ and $\scriptstyle [\cdot ]\,$ is the Iverson bracket.

The following identity holds

$\Pi _{\overline {\tau }}(n)={\frac {n!}{n^{\frac {\tau (n)}{2}}}}={\frac {(n-1)!}{n^{{\frac {\tau (n)}{2}}-1}}}\,$

where $\scriptstyle \tau (n)\,$ is the number of divisors of $\scriptstyle n\,$.

Properties

...

Sequences

A055067 Product of numbers < $n$ which do not divide $n$, $n$ ≥ 1, (or 1 if no such numbers).

{1, 1, 2, 3, 24, 20, 720, 630, 13440, 36288, 3628800, 277200, 479001600, 444787200, 5811886080, 20432412000, 20922789888000, 1097800704000, 6402373705728000, 304112751022080, ...}

See also

• Nondivisors
• Number of nondivisors
• Sum of nondivisors
• Nondivisorial or product of nondivisors

• Divisors
• Number of divisors
• Sum of divisors
• Divisorial or product of divisors

• Divisibility triangle

• Coprimorial and noncoprimorial