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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.
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