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%I #4 May 25 2018 21:59:15
%S 2,3,5,7,9,11,13,17,19,21,23,25,27,29,31,37,39,41,43,47,49,53,57,59,
%T 61,63,65,67,71,73,79,81,83,87,89,91,97,101,103,107,109,111,113,115,
%U 117,121,125,127,129,131,133,137,139,147,149,151,157,159,163,167,171
%N Heinz numbers of connected integer partitions with z-density -1.
%C First differs from A305078 at a(61) = 171, A305078(61) = 169.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph.
%C The z-density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(lcm(S)) where omega = A001221 is number of distinct prime factors.
%e 195 is the Heinz number of {2,3,6} with corresponding multiset multisystem {{1},{2},{1,2}}, which is connected with z-density -1.
%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%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 zens[n_]:=If[n==1,0,Total@Cases[FactorInteger[n],{p_,k_}:>k*(PrimeNu[PrimePi[p]]-1)]-PrimeNu[LCM@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]];
%t Select[Range[300],And[zens[#]==-1,Length[zsm[primeMS[#]]]==1]&]
%Y Cf. A001221, A056239, A112798, A286518, A302242, A303837, A304714, A304716, A305052, A305078, A305079.
%K nonn
%O 1,1
%A _Gus Wiseman_, May 25 2018
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