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Strong coprimality

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m
is strongly coprime to 
n
if and only if 
m
is coprime to 
n
and 
m
does not divide 
n  −  1
.

Strong coprimorial of n

The strong coprimorial (or strong phi-torial) of 
n
is the product of all positive integers 
i
up to and strongly coprime to 
n
. We say that 
i
is strongly coprime to 
n
if and only if 
i
is coprime to 
n
and 
i
does not divide 
n  −  1
.
where, for 
n = 0
and 
n = 1
, 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 divisors of 
n  −  1
and then multiply the residual together.
The strong coprimorial (or strong phi-torial) of 
n
is a divisor of the factorial of 
n
, since
where the second factor is the product of all positive integers up to and not strongly coprime to 
n
giving what might be called the strong noncoprimorial (or strong co-phi-torial) of 
n
. For example, with 
n = 8
, first we cull out 
2, 4, 6, 8,
next cull out the divisors of 
7
, finally multiply 
3, 5
giving 
15
.

Sequences

A181837
T (n, k)
= [[[Category:Pages using the math template without the tex argument|Strong coprimality]] 
k
is strongly prime to 
n
], 
n   ≥   0
, the indicator function of strong coprimality, triangle read by rows.
{0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, ...}
A181830 Number of positive integers up to and strongly coprime to 
n, n   ≥   0,
also called the strong totient of 
n
.
{0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 1, 6, 2, 6, 4, 4, 4, 11, 4, 12, 6, 6, 6, 18, 6, 12, 9, 14, 8, 22, 6, 22, 14, 14, 12, 20, 8, 27, 16, 20, 12, 32, 10, 34, 18, 18, 16, 42, 14, 32, 17, 26, 20, 46, 16, 32, 20, 28, 24, 54, 14, 48, 28, 32, 26, 41, 16, ...}
A181831 Sum of positive integers up to and strongly coprime to 
n, n   ≥   0
.
{1, 0, 0, 0, 3, 0, 9, 8, 12, 7, 37, 12, 50, 28, 36, 40, 105, 36, 132, 60, 84, 78, 217, 72, 190, 125, 201, 128, 350, 90, 393, 224, 267, 224, 366, 168, 575, 304, 408, 264, 730, 210, 807, 396, 456, 428, 1009, 336, 905, 443, ...}
A181832 Product of positive integers up to and strongly coprime to 
n, n   ≥   0,
also called the strong coprimorial (or strong phi-torial) of 
n
.
{1, 1, 1, 1, 1, 3, 1, 20, 15, 35, 7, 36288, 35, 277200, 1485, 4576, 9009, 20432412000, 5005, 1097800704000, 459459, 5912192, 2834325, 2322315553259520000, 1616615, 124672148625024, 4865140665, ...}
A181833 Number of positive integers up to and not strongly coprime to 
n, n   ≥   0,
also called the strong cototient of 
n
.
{0, 0, 2, 3, 4, 4, 6, 5, 6, 7, 9, 5, 10, 7, 10, 11, 12, 6, 14, 7, 14, 15, 16, 5, 18, 13, 17, 13, 20, 7, 24, 9, 18, 19, 22, 15, 28, 10, 22, 19, 28, 9, 32, 9, 26, 27, 30, 5, 34, 17, 33, 25, 32, 7, 38, 23, 36, 29, 34, 5, 46, ...}
A?????? Sum of positive integers up to and not strongly coprime to 
n, n   ≥   0
.
{?, ...}
A?????? Product of positive integers up to and not strongly coprime to 
n, n   ≥   0,
also called the strong noncoprimorial (or strong co-phi-torial) of 
n
.
{?, ...}

See also