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A316784 Number of orderless identity tree-factorizations of n.

%I A316784 #7 Nov 18 2018 19:26:12
%S A316784 1,1,1,1,1,2,1,2,1,2,1,4,1,2,2,3,1,4,1,4,2,2,1,10,1,2,2,4,1,8,1,6,2,2,
%T A316784 2,13,1,2,2,10,1,8,1,4,4,2,1,26,1,4,2,4,1,10,2,10,2,2,1,28,1,2,4,13,2,
%U A316784 8,1,4,2,8,1,46,1,2,4,4,2,8,1,26,3,2,1
%N A316784 Number of orderless identity tree-factorizations of n.
%C A316784 A factorization of n is a finite nonempty multiset of positive integers greater than 1 with product n. An orderless identity tree-factorization of n is either (case 1) the number n itself or (case 2) a finite set of two or more distinct orderless identity tree-factorizations, one of each factor in a factorization of n.
%C A316784 a(n) depends only on the prime signature of n. - _Andrew Howroyd_, Nov 18 2018
%H A316784 Andrew Howroyd, <a href="/A316784/b316784.txt">Table of n, a(n) for n = 1..10000</a>
%F A316784 a(p^n) = A300660(n) for prime p. - _Andrew Howroyd_, Nov 18 2018
%e A316784 The a(24)=10 orderless identity tree-factorizations:
%e A316784   24
%e A316784   (4*6)
%e A316784   (3*8)
%e A316784   (2*12)
%e A316784   (2*3*4)
%e A316784   (4*(2*3))
%e A316784   (3*(2*4))
%e A316784   (2*(2*6))
%e A316784   (2*(3*4))
%e A316784   (2*(2*(2*3)))
%t A316784 postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A316784 oltsfacs[n_]:=If[n<=1,{{}},Prepend[Select[Union@@Function[q,Sort/@Tuples[oltsfacs/@q]]/@DeleteCases[postfacs[n],{n}],UnsameQ@@#&],n]];
%t A316784 Table[Length[oltsfacs[n]],{n,100}]
%o A316784 (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(w[k], e)*v[i]))); w} \\ _Andrew Howroyd_, Nov 18 2018
%Y A316784 Cf. A001055, A001678, A004111, A292504, A292505, A295279, A300660, A316782.
%K A316784 nonn
%O A316784 1,6
%A A316784 _Gus Wiseman_, Jul 13 2018