%I #6 Dec 19 2017 02:40:26

%S 1,1,1,2,1,2,1,4,2,2,1,4,1,2,2,5,1,4,1,4,2,2,1,8,2,2,4,4,1,5,1,9,2,2,

%T 2,9,1,2,2,8,1,5,1,4,4,2,1,13,2,4,2,4,1,8,2,8,2,2,1,11,1,2,4,16,2,5,1,

%U 4,2,5,1,18,1,2,4,4,2,5,1,13,5,2,1,11,2

%N Number of twice-factorizations of n where the first factorization is strict and the latter factorizations are constant, i.e., type (P,Q,R).

%C a(n) is the number of ways to choose a perfect divisor of each factor in a strict factorization of n.

%F Dirichlet g.f.: Product_{n > 1}(1 + A089723(n)/n^s).

%e The a(24) = 8 twice-factorizations: (2)*(3)*(2*2), (2)*(3)*(4), (2)*(12), (3)*(2*2*2), (3)*(8), (2*2)*(6), (4)*(6), (24).

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

%t Table[Sum[Product[DivisorSigma[0,GCD@@FactorInteger[d][[All,2]]],{d,fac}],{fac,sfs[n]}],{n,100}]

%Y Cf. A001055, A005117, A045778, A063834, A089723, A279786, A281113, A296120, A296118.

%K nonn

%O 1,4

%A _Gus Wiseman_, Dec 05 2017