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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
![{\displaystyle {\frac {c_{n}!}{c_{n}\#}}\equiv \prod _{i=1}^{n}c_{i},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/ed21f7228e27cfb0c9734c146273a9e48e3c2aa7)
where
is the
th composite number.
Formulae for natural numbers
The compositorial of
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)}}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/174b5bf7b4c723d67cd373c9fef7e9c2097a9fbd)
where
is the factorial of
and
is the primorial of
.
The compositorial of
is the quotient of
by the squarefree kernel
(or radical
) of
![{\displaystyle {\frac {n!}{n\#}}={\frac {n!}{{\rm {rad}}(n!)}}.\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/ca399ad688413de70d6a4814f2ce62f2295e42ef)
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