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The
coprimorial of
is the
product of totatives of
(product of all
positive integers up to
and
coprime to
).
Also called “
phitorial” of
(
“phitorial”) or “
phitorial” of
(
“phitorial”) since the product involves
, the number of totatives of
.
Formulae
The
coprimorial of
is

φ! (n) = Πφ(n) := i = i = i [(i, n) =1], 
where
means
and
are
coprime and
[⋅] is the
Iverson bracket.
We have

where
is
Euler’s totient function,
is the
factorial of
,
[⋅] is the
Iverson bracket and
is the
Möbius function.
The
coprimorial (“phitorial”) of
and the
noncoprimorial (“
cophitorial”) of
are
divisors of the
factorial of
, since

Sequences
A001783 Coprimorial (product of
totatives) of
: product of numbers up to
that are coprime to
.

{1, 1, 2, 3, 24, 5, 720, 105, 2240, 189, 3628800, 385, 479001600, 19305, 896896, 2027025, 20922789888000, 85085, 6402373705728000, 8729721, 47297536000, 1249937325, ...} 
A066570 Noncoprimorial (product of
cototatives) of
: product of numbers up to
that have a prime factor in common with
. (
Empty product, i.e.
1, for
.)

{1, 2, 3, 8, 5, 144, 7, 384, 162, 19200, 11, 1244160, 13, 4515840, 1458000, 10321920, 17, 75246796800, 19, 278691840000, 1080203040, 899245670400, 23, 16686729658368000, 375000, ...}
A023896 Sum of
totatives of
: sum of numbers up to
that are coprime to
.

{1, 1, 3, 4, 10, 6, 21, 16, 27, 20, 55, 24, 78, 42, 60, 64, 136, 54, 171, 80, 126, 110, 253, 96, 250, 156, 243, 168, 406, 120, 465, 256, 330, 272, 420, 216, 666, 342, 468, 320, 820, 252, 903, 440, 540, ...}
A067392 Sum of
cototatives of
: sum of numbers up to
that have a prime factor in common with
. (
Empty sum, i.e.
0, for
.)

{0, 2, 3, 6, 5, 15, 7, 20, 18, 35, 11, 54, 13, 63, 60, 72, 17, 117, 19, 130, 105, 143, 23, 204, 75, 195, 135, 238, 29, 345, 31, 272, 231, 323, 210, 450, 37, 399, 312, 500, 41, 651, 43, 550, 495, 575, 47, ...}
See also