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

1,2

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.

Links

Table of n, a(n) for n=1..66.

Example

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}}

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&]

Crossrefs

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

Sequence in context: A360550 A326621 A324758 * A316413 A360009 A359905

Adjacent sequences: A305501 A305502 A305503 * A305505 A305506 A305507

Keyword

nonn

Author

Gus Wiseman, Jun 03 2018

Status

approved