Noncoprimorial

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

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

Formulae

The noncoprimorial of $\scriptstyle n\,$ is thus

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

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

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

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

Sequences

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

See also

• Coprimorial