(Redirected from Product of nondivisors)
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The nondivisorial of
is the product of nondivisors of
(product of all positive integers up to
which are not factors of
).
By analogy with phi-torial (phitorial) for coprimorial, might be called co-tau-torial of
(
co-tau-torial) or co-tautorial of
(
co-tautorial) since the product involves
numbers, where
is the number of nondivisors of
.
Formulae
Thus the nondivisorial of
is
![{\displaystyle \Pi _{\overline {\tau }}(n)\equiv \prod _{\stackrel {i=1}{i\nmid n}}^{n}i=\prod _{i=1}^{n}i^{(1-[n{\bmod {i}}\,=\,0])}={\frac {n!}{\Pi _{\tau }(n)}}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/768e215eb341131cb26b63204097edd2068fdfc4)
where
is the number of nondivisors of
and
is the Iverson bracket.
The following identity holds
![{\displaystyle \Pi _{\overline {\tau }}(n)={\frac {n!}{n^{\frac {\tau (n)}{2}}}}={\frac {(n-1)!}{n^{{\frac {\tau (n)}{2}}-1}}}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/5f79b89961c5690fe7814b43845233d4250495a1)
where
is the number of divisors of
.
Properties
...
Sequences
A055067 Product of numbers <
which do not divide
,
≥ 1, (or 1 if no such numbers).
- {1, 1, 2, 3, 24, 20, 720, 630, 13440, 36288, 3628800, 277200, 479001600, 444787200, 5811886080, 20432412000, 20922789888000, 1097800704000, 6402373705728000, 304112751022080, ...}
See also