A305504 Heinz numbers of integer partitions whose distinct parts plus 1 are connected.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 22, 23, 25, 27, 29, 31, 32, 33, 34, 37, 40, 41, 43, 44, 46, 47, 49, 50, 53, 55, 57, 59, 61, 62, 64, 66, 67, 68, 71, 73, 79, 80, 81, 82, 83, 85, 88, 89, 92, 93, 94, 97, 99, 100, 101, 103, 107, 109, 110, 113, 115

Offset: 1

Author

Gus Wiseman, Jun 03 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 greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A partition y is said to be connected if G(U(y + 1)) is a connected graph, where U(y + 1) is the set of distinct successors of the parts of y.

This is intended to be a cleaner form of A305078, where the treatment of empty multisets is arbitrary.

Examples

			The sequence of entries together with the corresponding twice-prime-factored multiset partitions (see A275024) begins:
   1: {}
   2: {{1}}
   3: {{2}}
   4: {{1},{1}}
   5: {{1,1}}
   7: {{3}}
   8: {{1},{1},{1}}
   9: {{2},{2}}
  10: {{1},{1,1}}
  11: {{1,2}}
  13: {{4}}
  16: {{1},{1},{1},{1}}
  17: {{1,1,1}}
  19: {{2,2}}
  20: {{1},{1},{1,1}}
  22: {{1},{1,2}}
		

Crossrefs

Cf. A001221, A048143, A056239, A112798, A275024, A286518, A290103, A302242, A304714, A304716, A305052, A305078, A305079, A305501.

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]]]]]]]]];
Select[Range[300],Length[zsm[primeMS[#]+1]]<=1&]