A295935 - OEIS

%I #7 Dec 19 2017 02:39:22

%S 1,1,1,3,1,2,1,5,3,2,1,5,1,2,2,12,1,5,1,5,2,2,1,10,3,2,5,5,1,5,1,18,2,

%T 2,2,15,1,2,2,10,1,5,1,5,5,2,1,22,3,5,2,5,1,10,2,10,2,2,1,13,1,2,5,40,

%U 2,5,1,5,2,5,1,28,1,2,5,5,2,5,1,22,12,2,1

%N Number of twice-factorizations of n where the latter factorizations are constant, i.e., type (P,P,R).

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

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

%e The a(24) = 10 twice-factorizations are:

%e (2)*(2)*(2)*(3), (2)*(3)*(2*2), (3)*(2*2*2)

%e (2)*(2)*(6), (2*2)*(6),

%e (2)*(3)*(4),

%e (2)*(12),

%e (3)*(8),

%e (4)*(6),

%e (24).

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

%t Table[Sum[Product[Length[Divisors[GCD@@FactorInteger[d][[All,2]]]],{d,f}],{f,facs[n]}],{n,100}]

%Y Cf. A000005, A001055, A052409, A052410, A089723, A279784, A281113, A284639, A295923, A295924, A295931.

%K nonn

%O 1,4

%A _Gus Wiseman_, Nov 29 2017