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# Noncoprimorial

The noncoprimorial of ${\displaystyle \scriptstyle n\,}$ is the product of cototatives of ${\displaystyle \scriptstyle n\,}$ (product of all positive integers up to ${\displaystyle \scriptstyle n\,}$ and noncoprime to ${\displaystyle \scriptstyle n\,}$, i.e. not coprime to ${\displaystyle \scriptstyle n\,}$.)

By analogy with phi-torial (phitorial) for coprimorial, might be called co-phi-torial of ${\displaystyle \scriptstyle n\,}$ (${\displaystyle \scriptstyle n\,}$ co-phi-torial) or co-phitorial of ${\displaystyle \scriptstyle n\,}$ (${\displaystyle \scriptstyle n\,}$ co-phitorial) since the product involves ${\displaystyle \scriptstyle {\overline {\varphi }}(n)\,}$ numbers, where ${\displaystyle \scriptstyle {\overline {\varphi }}(n)\,}$ is Euler's cototient function.

## Formulae

The noncoprimorial of ${\displaystyle \scriptstyle n\,}$ is thus

${\displaystyle {\overline {\varphi }}_{_{_{!}}}(n)=\Pi _{\overline {\varphi }}(n)\equiv \prod _{\stackrel {i=1}{i\not \perp n}}^{n}i=\prod _{\stackrel {i=1}{(i,n)\neq 1}}^{n}i=\prod _{i=1}^{n}i^{[(i,n)\neq 1]}\,}$

where ${\displaystyle \scriptstyle i\not \perp n\,}$ means ${\displaystyle \scriptstyle i\,}$ and ${\displaystyle \scriptstyle n\,}$ are nonorthogonal numbers (i.e. noncoprime) and ${\displaystyle \scriptstyle [\cdot ]\,}$ is Iverson bracket.

The coprimorial (phi-torial) of ${\displaystyle \scriptstyle n\,}$ and the noncoprimorial (co-phi-torial) of ${\displaystyle \scriptstyle n\,}$ are divisors of the factorial of n.

${\displaystyle n!=\varphi _{_{_{!}}}(n)~{\overline {\varphi }}_{_{_{!}}}(n).\,}$

## Sequences

Noncoprimorial (product of cototatives) of ${\displaystyle \scriptstyle n\,}$: product of numbers ${\displaystyle \scriptstyle \leq \,n\,}$ that have a prime factor in common with ${\displaystyle \scriptstyle n,\ n\,\geq \,1,\,}$ (Cf. A066570) (empty product, 1, for ${\displaystyle \scriptstyle n=1\,}$) gives

{1, 2, 3, 8, 5, 144, 7, 384, 162, 19200, 11, 1244160, 13, 4515840, 1458000, 10321920, 17, 75246796800, 19, 278691840000, 1080203040, 899245670400, 23, 16686729658368000, 375000, 663152807116800, ...}