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Weak coprimality

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m
is weakly prime to
n
if and only if
m
is coprime to
n
and
m
divides
n  −  1
.

Weak coprimorial of n

The weak coprimorial (or weak phi-torial) of
n
is the product of all positive integers
i
up to and weakly coprime to
n
. We say that
i
is weakly coprime to
n
if and only if
i
is coprime to
n
and
i
divides
n  −  1
.
ϕ!(n)  :=



n
i  = 1
i  ∣  (n  − 1)
in
  
i, n ≥ 0,
where, for
n = 0
, we get the empty product (defined as the multiplicative identity, i.e. 1). We take the positive integers below
n
, first cull out those
i
that have prime factors in common with
n
, next cull out those
i
which are not divisors of
n  −  1
and then multiply the residual together.
The weak coprimorial (or weak phi-torial) of
n
is a divisor of the factorial of
n
, since
n!  =  ϕ!(n) ϕ!(n),
where the second factor is the product of all positive integers up to and not weakly coprime to
n
giving what might be called the weak noncoprimorial (or weak co-phi-torial) of
n
.

Sequences

A??????
T  (n, k) =
[
k
is weakly prime to
n
],
n   ≥   0
, the indicator function of weak coprimality, triangle read by rows.
{?, ...}
A?????? Number of positive integers up to and weakly coprime to
n, n   ≥   0,
also called the weak totient of
n
.
{?, ...}
A?????? Sum of positive integers up to and weakly coprime to
n, n   ≥   0
.
{?, ...}
A?????? Product of positive integers up to and weakly coprime to
n, n   ≥   0,
also called the weak coprimorial (or weak phi-torial) of
n
.
{?, ...}
A?????? Number of positive integers up to and not weakly coprime to
n, n   ≥   0,
also called the weak cototient of
n
.
{?, ...}
A?????? Sum of positive integers up to and not weakly coprime to
n, n   ≥   0
.
{?, ...}
A?????? Product of positive integers up to and not weakly coprime to
n, n   ≥   0,
also called the weak noncoprimorial (or weak co-phi-torial) of
n
.
{?, ...}

See also