A305081 - OEIS

%I #5 May 25 2018 21:59:22

%S 2,3,5,7,9,11,13,17,19,23,25,27,29,31,37,41,43,47,49,53,59,61,67,71,

%T 73,79,81,83,89,91,97,101,103,107,109,113,121,125,127,131,137,139,149,

%U 151,157,163,167,173,179,181,191,193,197,199,203,211,223,227,229

%N Heinz numbers of z-trees. Heinz numbers of connected integer partitions with pairwise indivisible parts and z-density -1.

%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 clutter density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(lcm(S)) where omega = A001221.

%e 4331 is the Heinz number of {18,20}, which is a z-tree corresponding to the multiset multisystem {{1,2,2},{1,1,3}}.

%e 17927 is the Heinz number of {4,6,45}, which is a z-tree corresponding to the multiset multisystem {{1,1},{1,2},{2,2,3}}.

%e 27391 is the Heinz number of {4,4,6,14}, which is a z-tree corresponding to the multiset multisystem {{1,1},{1,1},{1,2},{1,4}}.

%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,Select[Tuples[primeMS[#],2],UnsameQ@@#&&Divisible@@#&]=={}]&]

%Y Cf. A001221, A030019, A056239, A112798, A286520, A302242, A303362, A303837, A304118, A304714, A304716, A305052, A305078, A305079.

%K nonn

%O 1,1

%A _Gus Wiseman_, May 25 2018