(Redirected from Phitorial)
There are no approved revisions of this page, so it may
not have been
reviewed.
This article needs more work.
Please help by expanding it!
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