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Number of twice-factorizations of n of type (Q,R,Q).
4

%I #8 Dec 19 2017 02:40:41

%S 1,1,1,1,1,2,1,2,1,2,1,3,1,2,2,2,1,3,1,3,2,2,1,5,1,2,2,3,1,5,1,3,2,2,

%T 2,7,1,2,2,5,1,5,1,3,3,2,1,7,1,3,2,3,1,5,2,5,2,2,1,9,1,2,3,6,2,5,1,3,

%U 2,5,1,9,1,2,3,3,2,5,1,7,2,2,1,9,2,2,2

%N Number of twice-factorizations of n of type (Q,R,Q).

%e The a(36) = 7 twice-factorizations are (2*3)*(6), (6)*(2*3), (2*3*6), (2*18), (3*12), (4*9), (36).

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

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t Table[Length[Join@@Function[fac,Select[Join@@Permutations/@sps[fac],SameQ@@Times@@@#&]]/@sfs[n]],{n,100}]

%Y Cf. A000005, A001055, A005117, A045778, A052409, A052410, A089723, A279790, A279791, A284639, A292886, A296132.

%K nonn

%O 1,6

%A _Gus Wiseman_, Dec 05 2017