A323090 - OEIS

%I #5 Jan 04 2019 17:33:16

%S 1,1,1,0,1,2,1,0,0,2,1,2,1,2,2,0,1,2,1,2,2,2,1,2,0,2,0,2,1,5,1,0,2,2,

%T 2,3,1,2,2,2,1,5,1,2,2,2,1,2,0,2,2,2,1,2,2,2,2,2,1,7,1,2,2,0,2,5,1,2,

%U 2,5,1,4,1,2,2,2,2,5,1,2,0,2,1,7,2,2,2

%N Number of strict factorizations of n using elements of A007916 (numbers that are not perfect powers).

%e The a(72) = 4 factorizations are (2*3*12), (3*24), (6*12), (72). Missing from this list and not strict are (2*2*2*3*3), (2*2*3*6), (2*6*6), (2*2*18), while missing from the list and using perfect powers are (2*36), (2*4*9), (3*4*6), (4*18), (8*9).

%t radQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]==1];

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

%t Table[Length[facssr[n]],{n,100}]

%Y Positions of 0's are A246547.

%Y Positions of 1's are A000040.

%Y Positions of 2's are A084227.

%Y Positions of 3's are A085986.

%Y Positions of 4's are A143610.

%Y Cf. A001055, A001597, A007916, A025147, A045778, A052410, A303707, A323054, A323087, A323088, A323089.

%K nonn

%O 1,6

%A _Gus Wiseman_, Jan 04 2019