A305761 - OEIS

%I #5 Jul 15 2018 13:24:05

%S 91,203,247,299,301,377,427,551,553,559,611,689,703,707,791,817,851,

%T 923,949,973,1027,1073,1081,1141,1159,1247,1267,1313,1339,1349,1363,

%U 1391,1393,1501,1537,1591,1603,1679,1703,1739,1757,1769,1781,1807,1897,1919,1961

%N Nonprime Heinz numbers of z-trees.

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). 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. 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. The z-density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(lcm(S)). Finally, a z-tree of weight n is a connected strict integer partition of n with at least two pairwise indivisible parts and z-density -1.

%e 2639 is the Heinz number of {4,6,10}, a z-tree corresponding to the multiset system {{1,1},{1,2},{1,3}}.

%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 zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];

%t Select[Range[3000],With[{p=primeMS[#]},And[UnsameQ@@p,Length[p]>1,zensity[p]==-1,Length[zsm[p]]==1,Select[Tuples[p,2],UnsameQ@@#&&Divisible@@#&]=={}]]&]

%Y Cf. A030019, A056239, A112798, A286520, A302242, A303362, A303837, A304118, A304714, A304716, A305052, A305078, A305079, A305081.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jun 10 2018