Compositorial

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The $\scriptstyle n\,$th compositorial number, denoted $\scriptstyle {\frac {c_{n}!}{c_{n}\#}}\,$, is defined as the product of the first $\scriptstyle n\,$ composites, the 0^th compositorial number being the empty product, defined as the multiplicative identity, i.e. 1.

The compositorial of a natural number $\scriptstyle n\,$, denoted $\scriptstyle {\frac {n!}{n\#}}\,$, is the product of all positive composite integers up to $\scriptstyle n\,$, the compositorial of 0 being the empty product, defined as the multiplicative identity, i.e. 1.

Formulae

Formulae for composite numbers

The $\scriptstyle n\,$^th compositorial number is given by

${\frac {c_{n}!}{c_{n}\#}}\equiv \prod _{i=1}^{n}c_{i},\,$

where $\scriptstyle c_{i}\,$ is the $\scriptstyle i\,$th composite number.

Formulae for natural numbers

The compositorial of $\scriptstyle n\,$ is given by

${\frac {n!}{n\#}}\equiv \prod _{i=1}^{n}i^{\chi _{\{{\rm {composites\}}}}(i)}={\frac {n!}{\prod _{i=1}^{n}i^{\chi _{\{{\rm {primes\}}}}(i)}}},\,$

where $\scriptstyle n!\,$ is the factorial of $\scriptstyle n\,$ and $\scriptstyle n\#\,$ is the primorial of $\scriptstyle n\,$.

The compositorial of $\scriptstyle n\,$ is the quotient of $\scriptstyle n!\,$ by the squarefree kernel $\scriptstyle {\rm {sqf}}(n!)\,$ (or radical $\scriptstyle {\rm {rad}}(n!)\,$) of $\scriptstyle n!\,$

${\frac {n!}{n\#}}={\frac {n!}{{\rm {rad}}(n!)}}.\,$

Sequences

A036691 The compositorial numbers, $\scriptstyle {\frac {c_{n}!}{c_{n}\#}},\ n\,\geq \,0\,$.

{1, 4, 24, 192, 1728, 17280, 207360, 2903040, 43545600, 696729600, 12541132800, 250822656000, 5267275776000, 115880067072000, 2781121609728000, 69528040243200000, 1807729046323200000, ...}

The compositorial of $\scriptstyle n\,$, i.e. $\scriptstyle {\frac {n!}{n\#}},\ n\,\geq \,0\,$. (A049614, $\scriptstyle n\,\geq \,1.\,$)

{1, 1, 1, 1, 4, 4, 24, 24, 192, 1728, 17280, 17280, 207360, 207360, 2903040, 43545600, 696729600, 696729600, 12541132800, 12541132800, 250822656000, 5267275776000, 115880067072000, ...}

See also

• A007947 Largest squarefree number dividing $\scriptstyle n\,$ (the squarefree kernel, or radical, of $\scriptstyle n\,$).

• Factorial
• Primorial