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# Compositorial

The th compositorial number, denoted , is defined as the product of the first composites, the 0^{ th} compositorial number being the empty product, defined as the multiplicative identity, i.e. 1.

The compositorial of a natural number , denoted , is the product of all positive composite integers up to , the compositorial of 0 being the empty product, defined as the multiplicative identity, i.e. 1.

## Contents

## Formulae

### Formulae for composite numbers

The ^{th} compositorial number is given by

where is the th composite number.

### Formulae for natural numbers

The compositorial of is given by

where is the factorial of and is the primorial of .

The compositorial of is the quotient of by the squarefree kernel (or radical ) of

## Sequences

A036691 The compositorial numbers, .

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

The compositorial of , i.e. . (A049614, )

- {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 (the squarefree kernel, or radical, of ).