A303707 Number of factorizations of n using elements of A007916 (numbers that are not perfect powers).

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 5, 1, 1, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 1, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 1, 2, 1, 9, 2, 2, 2

Offset: 1

Author

Gus Wiseman, Apr 29 2018

Keywords

• nonn

Comments

First differs from A081707 at a(60) = 9, A081707(60) = 8.

Examples

			The a(60) = 9 factorizations are (2*2*3*5), (2*2*15), (2*3*10), (2*5*6), (2*30), (3*20), (5*12), (6*10), (60).
		

Crossrefs

Cf. A000837, A001055, A001597, A007716, A007916, A045778, A052409, A052410, A162247, A281113, A281116, A303708, A303709, A303710.

Programs

Mathematica

radQ[n_]:=Or[n===1,GCD@@FactorInteger[n][[All,2]]===1];
facsr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsr[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],radQ]}]];
Table[Length[facsr[n]],{n,100}]

Formula

Dirichlet g.f.: Product_{n in A007916} 1/(1 - n^s).