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# Coprimorial

(Redirected from Phi-torial)

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

## Formulae

The coprimorial of
 n
is
φ! (n)  =  Πφ(n)  :=
 n ∏ i   = 1i ⊥n

i  =
 n ∏ i   = 1(i, n)  = 1

i  =
 n ∏ i   = 1

i [(i, n) =1],
where
 i ⊥ n
means
 i
and
 n
are coprime and [⋅] is the Iverson bracket.

We have

 φ! (n) n φ (n)
=
 Πφ(n) n φ (n)
=
 d ∣n ∏ d ∣n

 d ! d  d
μ (
 n d
)
,
where
 φ (n)
is Euler’s totient function,
 n!
is the factorial of
 n
, [⋅] is the Iverson bracket and
 μ (n)
is the Möbius function.
The coprimorial (“phi-torial”) of
 n
and the noncoprimorial (“co-phi-torial”) of
 n
are divisors of the factorial of
 n
, since
 n!  =  φ!(n) x̅φ!(n).

## Sequences

A001783 Coprimorial (product of totatives) of
 n
: product of numbers up to
 n
that are coprime to
 n, n   ≥   1
.
 {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
 n
: product of numbers up to
 n
that have a prime factor in common with
 n, n   ≥   1
. (Empty product, i.e. 1, for
 n = 1
.)
{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
 n
: sum of numbers up to
 n
that are coprime to
 n, n   ≥   1
.
{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
 n
: sum of numbers up to
 n
that have a prime factor in common with
 n, n   ≥   1
. (Empty sum, i.e. 0, for
 n = 1
.)
{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, ...}