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%I #8 Nov 02 2018 11:23:48
%S 0,1,1,2,1,1,1,3,2,1,1,2,1,1,1,5,1,2,1,2,1,1,1,4,2,1,3,2,1,1,1,7,1,1,
%T 1,4,1,1,1,4,1,1,1,2,2,1,1,7,2,2,1,2,1,4,1,4,1,1,1,3,1,1,2,11,1,1,1,2,
%U 1,1,1,8,1,1,2,2,1,1,1,7,5,1,1,3,1,1,1
%N Number of connected factorizations of n into positive integers > 1 with z-density -1.
%C These are z-trees (A303837, A305081, A305253, A321279) where we relax the requirement of pairwise indivisibility.
%C Given a finite multiset S of positive integers greater than 1, let G(S) be the simple labeled graph with vertices the distinct elements of S and with edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. Then S is said to be connected if G(S) is a connected graph.
%C The z-density of a factorization S is defined to be Sum_{s in S} (omega(s) - 1) - omega(n), where omega = A001221 and n is the product of S.
%e The a(72) = 8 factorizations are (2*2*3*6), (2*2*18), (2*3*12), (2*36), (3*4*6), (3*24), (4*18), (72). Missing from this list but still connected are (2*6*6),(6*12).
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
%t zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[Times@@s];
%t Table[Length[Select[facs[n],And[zensity[#]==-1,Length[zsm[#]]==1]&]],{n,100}]
%Y Cf. A001055, A001221, A030019, A286518, A303837, A304118, A304382, A305052, A305081, A305193, A305253, A319786, A321229, A321253.
%K nonn
%O 1,4
%A _Gus Wiseman_, Nov 01 2018
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