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Number of non-constant factorizations of n whose distinct factors are pairwise coprime.
2

%I #7 Sep 23 2019 13:43:57

%S 0,0,0,0,0,1,0,0,0,1,0,2,0,1,1,0,0,2,0,2,1,1,0,2,0,1,0,2,0,4,0,0,1,1,

%T 1,4,0,1,1,2,0,4,0,2,2,1,0,3,0,2,1,2,0,2,1,2,1,1,0,6,0,1,2,0,1,4,0,2,

%U 1,4,0,4,0,1,2,2,1,4,0,3,0,1,0,6,1,1,1

%N Number of non-constant factorizations of n whose distinct factors are pairwise coprime.

%F a(n) = A327399(n) - A089723(n).

%e The factorizations of 6, 12, 30, 48, 60, 180, and 210:

%e (2*3) (3*4) (5*6) (3*16) (3*20) (4*45) (3*70)

%e (2*2*3) (2*15) (3*4*4) (4*15) (5*36) (5*42)

%e (3*10) (2*2*2*2*3) (5*12) (9*20) (6*35)

%e (2*3*5) (3*4*5) (4*5*9) (7*30)

%e (2*2*15) (5*6*6) (10*21)

%e (2*2*3*5) (2*2*45) (14*15)

%e (3*3*20) (2*105)

%e (2*2*5*9) (5*6*7)

%e (3*3*4*5) (2*3*35)

%e (2*2*3*3*5) (2*5*21)

%e (2*7*15)

%e (3*5*14)

%e (3*7*10)

%e (2*3*5*7)

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Table[Length[Select[facs[n],CoprimeQ@@Union[#]&]],{n,100}]

%Y Factorizations that are constant or whose distinct parts are pairwise coprime are counted by A327399.

%Y Numbers with pairwise coprime distinct prime indices are A304711.

%Y Cf. A001055, A089723, A281116, A318721, A302569, A319269, A327407, A327517.

%K nonn

%O 1,12

%A _Gus Wiseman_, Sep 22 2019