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Compositorial

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

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

Formulae

Formulae for composite numbers

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

${\frac {c_{n}!}{c_{n}\#}}\equiv \prod _{i=1}^{n}c_{i},\,$ where $c_{i}\,$ is the $i\,$ th composite number.

Formulae for natural numbers

The compositorial of $n\,$ 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 $n!\,$ is the factorial of $n\,$ and $n\#\,$ is the primorial of $n\,$ .

The compositorial of $n\,$ is the quotient of $n!\,$ by the squarefree kernel ${\rm {sqf}}(n!)\,$ (or radical ${\rm {rad}}(n!)\,$ ) of $n!\,$ ${\frac {n!}{n\#}}={\frac {n!}{{\rm {rad}}(n!)}}.\,$ Sequences

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

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

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

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

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