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The noncoprimorial of is the product of cototatives of (product of all positive integers up to and noncoprime to , i.e. not coprime to .)

By analogy with phi-torial (phitorial) for coprimorial, might be called co-phi-torial of ( co-phi-torial) or co-phitorial of ( co-phitorial) since the product involves numbers, where is Euler's cototient function.


The noncoprimorial of is thus

where means and are nonorthogonal numbers (i.e. noncoprime) and is Iverson bracket.

The coprimorial (phi-torial) of and the noncoprimorial (co-phi-torial) of are divisors of the factorial of n.


Noncoprimorial (product of cototatives) of : product of numbers that have a prime factor in common with (Cf. A066570) (empty product, 1, for ) gives

{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, 663152807116800, ...}

See also