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

1,6

Comments

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

Links

Table of n, a(n) for n=1..87.

Formula

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

Example

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).

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}]

Crossrefs

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

Sequence in context: A326516 A081707 A368414 * A335521 A323087 A321747

Adjacent sequences: A303704 A303705 A303706 * A303708 A303709 A303710

Keyword

nonn

Author

Gus Wiseman, Apr 29 2018

Status

approved