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Coprimorial

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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   = 1
i ⊥n
  
i  = 

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

We have

φ! (n)
nφ (n)
 = 
Πφ(n)
nφ (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, ...}

See also