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%I #5 Sep 12 2018 15:10:42
%S 0,1,1,1,1,2,1,3,1,2,1,9,1,2,2,11,1,9,1,9,2,2,1,44,1,2,3,9,1,18,1,45,
%T 2,2,2,66,1,2,2,44,1,18,1,9,9,2,1,225,1,9,2,9,1,44,2,44,2,2,1,132,1,2,
%U 9,197,2,18,1,9,2,18,1,450,1,2,9,9,2,18,1,225
%N Number of complete planar branching factorizations of n.
%C A planar branching factorization of n is either the number n itself or a sequence of at least two planar branching factorizations, one of each factor in an ordered factorization of n. A planar branching factorization is complete if the leaves are all prime numbers.
%F a(prime^n) = A001003(n - 1).
%F a(product of n distinct primes) = A032037(n).
%e The a(12) = 9 trees:
%e (2*2*3), (2*3*2), (3*2*2),
%e (2*(2*3)), (2*(3*2)), (3*(2*2)), ((2*2)*3), ((2*3)*2), ((3*2)*2).
%t ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]]
%t otfs[n_]:=Prepend[Join@@Table[Tuples[otfs/@f],{f,Select[ordfacs[n],Length[#]>1&]}],n];
%t Table[Length[Select[otfs[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,100}]
%Y Cf. A000108, A001055, A007716, A074206, A118376, A277130, A281118, A281119, A292505, A317144, A319122, A319138.
%K nonn
%O 1,6
%A _Gus Wiseman_, Sep 11 2018
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