%I #8 Oct 26 2019 15:33:44

%S 1,2,3,9,7,17,15,40,39,56,56,126,101,165,197,336,297,496,490,774,837,

%T 1114,1255,1948,2007,2638,3127,4123,4565,6201,6842,9131,10311,12904,

%U 14988,19516,21637,26995,31488,39250,44583,55418,63261,77683,89935,108068,124754

%N Number of ways to choose an integer partition of each factor in a factorization of n.

%F Dirichlet g.f.: Product_{n > 1} 1 / (1 - P(n) / n^s) where P = A000041. [clarified by _Ilya Gutkovskiy_, Oct 26 2019]

%e The a(4) = 9 ways: (1+1)*(1+1), (1+1+1+1), (1+1)*(2), (2)*(1+1), (2+1+1), (2)*(2), (2+2), (3+1), (4).

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

%t Table[Sum[Times@@PartitionsP/@fac,{fac,facs[n]}],{n,10}]

%Y Cf. A000041, A001055, A001970, A063834, A065026, A066739, A066815, A121229, A281113, A284639, A318949.

%K nonn

%O 1,2

%A _Gus Wiseman_, Sep 05 2018