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is weakly prime to
if and only if
is
coprime to
and
divides
.
Weak coprimorial of n
The
weak coprimorial (or
weak phi-torial) of
is the product of all positive integers
up to and
weakly coprime to
. We say that
is weakly coprime to
if and only if
is
coprime to
and
divides
.
-
where, for
, we get the
empty product (defined as the
multiplicative identity, i.e.
1). We take the positive integers below
, first cull out those
that have
prime factors in common with
, next cull out those
which are not divisors of
and then multiply the residual together.
The weak coprimorial (or
weak phi-torial) of
is a divisor of the
factorial of
, since
-
where the second factor is the product of all positive integers up to and
not weakly coprime to
giving what might be called the
weak noncoprimorial (or
weak co-phi-torial) of
.
Sequences
A??????
[
is
weakly prime to
],
, the indicator function of weak coprimality, triangle read by rows.
-
{?, ...}
A?????? Number of positive integers up to and
weakly coprime to
also called the
weak totient of
.
-
{?, ...}
A?????? Sum of positive integers up to and
weakly coprime to
.
-
{?, ...}
A?????? Product of positive integers up to and
weakly coprime to
also called the
weak coprimorial (or
weak phi-torial) of
.
-
{?, ...}
A?????? Number of positive integers up to and
not weakly coprime to
also called the
weak cototient of
.
-
{?, ...}
A?????? Sum of positive integers up to and
not weakly coprime to
.
-
{?, ...}
A?????? Product of positive integers up to and
not weakly coprime to
also called the
weak noncoprimorial (or
weak co-phi-torial) of
.
-
{?, ...}
See also
