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# Compositorial

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 0th 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, $\scriptstyle \frac{c_n!}{c_n\#},\ n \,\ge\, 0 \,$.

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

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

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