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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 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.

Contents

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, ...}

See also


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