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%I #15 Nov 18 2018 19:26:30
%S 1,1,1,2,1,2,1,4,2,2,1,6,1,2,2,11,1,6,1,6,2,2,1,20,2,2,4,6,1,8,1,30,2,
%T 2,2,27,1,2,2,20,1,8,1,6,6,2,1,74,2,6,2,6,1,20,2,20,2,2,1,38,1,2,6,96,
%U 2,8,1,6,2,8,1,114,1,2,6,6,2,8,1,74,11,2,1
%N Number of orderless tree-factorizations of n.
%C A factorization of n is a finite multiset of positive integers greater than 1 with product n. An orderless tree-factorization of n is either (case 1) the number n itself or (case 2) a finite multiset of two or more orderless tree-factorizations, one of each factor in a factorization of n.
%C a(n) depends only on the prime signature of n. - _Andrew Howroyd_, Nov 18 2018
%H Michael De Vlieger, <a href="/A292504/b292504.txt">Table of n, a(n) for n = 1..16384</a>
%H Michael De Vlieger, <a href="/A292504/a292504.txt">Records and indices of records</a>.
%F a(p^n) = A141268(n) for prime p. - _Andrew Howroyd_, Nov 18 2018
%e The a(16)=11 orderless tree-factorizations are: 16, (28), (2(24)), (2(2(22))), (2(222)), (44), (4(22)), ((22)(22)), (224), (22(22)), (2222).
%t postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t oltfacs[n_]:=If[n<=1,{{}},Prepend[Union@@Function[q,Sort/@Tuples[oltfacs/@q]]/@DeleteCases[postfacs[n],{n}],n]];
%t Table[Length[oltfacs[n]],{n,83}]
%o (PARI) seq(n)={my(v=vector(n), w=vector(n)); w[1]=v[1]=1; for(k=2, n, w[k]=v[k]+1; forstep(j=n\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j] += binomial(e+w[k]-1, e)*v[i]))); w} \\ _Andrew Howroyd_, Nov 18 2018
%Y Cf. A000311, A001055, A050336, A141268, A281118, A292505.
%K nonn
%O 1,4
%A _Gus Wiseman_, Sep 17 2017
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