A305761 Nonprime Heinz numbers of z-trees.

91, 203, 247, 299, 301, 377, 427, 551, 553, 559, 611, 689, 703, 707, 791, 817, 851, 923, 949, 973, 1027, 1073, 1081, 1141, 1159, 1247, 1267, 1313, 1339, 1349, 1363, 1391, 1393, 1501, 1537, 1591, 1603, 1679, 1703, 1739, 1757, 1769, 1781, 1807, 1897, 1919, 1961

Offset: 1

Author

Gus Wiseman, Jun 10 2018

Keywords

• nonn

Comments

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.

Examples

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

Crossrefs

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

Programs

Mathematica

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
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@@#&]=={}]]&]