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# Compositorial

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

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

## Formulae

### Formulae for composite numbers

The ${\displaystyle \scriptstyle n\,}$th compositorial number is given by

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

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

### Formulae for natural numbers

The compositorial of ${\displaystyle \scriptstyle n\,}$ is given by

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

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

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

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

## Sequences

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

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

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

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