A296120 - OEIS

%I #6 Dec 19 2017 02:39:50

%S 1,1,1,1,1,3,1,3,1,3,1,6,1,3,3,4,1,6,1,6,3,3,1,13,1,3,3,6,1,12,1,7,3,

%T 3,3,14,1,3,3,13,1,12,1,6,6,3,1,25,1,6,3,6,1,13,3,13,3,3,1,31,1,3,6,

%U 11,3,12,1,6,3,12,1,36,1,3,6,6,3,12,1,25,4,3

%N Number of ways to choose a strict factorization of each factor in a strict factorization of n.

%F Dirichlet g.f.: Product_{n > 1}(1 + A045778(n)/n^s).

%e The a(36) = 14 twice-factorizations:

%e (36), (4*9), (3*12), (2*18), (2*3*6),

%e (4)*(9), (3)*(12), (3)*(3*4), (3)*(2*6), (2)*(18), (2)*(3*6), (2)*(2*9),

%e (2)*(3)*(6), (2)*(3)*(2*3).

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

%t Table[Sum[Times@@Length/@sfs/@fac,{fac,sfs[n]}],{n,100}]

%Y Cf. A000009, A005117, A045778, A050345, A279785, A281113, A294788, A296118, A296119, A296121.

%K nonn

%O 1,6

%A _Gus Wiseman_, Dec 05 2017