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A305078 Heinz numbers of connected integer partitions. 61
2, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 129, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Given a finite multiset 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. This sequence lists all Heinz numbers of multisets S such that G(S) is a connected graph.

LINKS

Table of n, a(n) for n=1..60.

EXAMPLE

The sequence of all connected multiset multisystems (see A302242, A112798) begins:

   2: {{}}

   3: {{1}}

   5: {{2}}

   7: {{1,1}}

   9: {{1},{1}}

  11: {{3}}

  13: {{1,2}}

  17: {{4}}

  19: {{1,1,1}}

  21: {{1},{1,1}}

  23: {{2,2}}

  25: {{2},{2}}

  27: {{1},{1},{1}}

  29: {{1,3}}

  31: {{5}}

  37: {{1,1,2}}

  39: {{1},{1,2}}

  41: {{6}}

  43: {{1,4}}

  47: {{2,3}}

  49: {{1,1},{1,1}}

  53: {{1,1,1,1}}

  57: {{1},{1,1,1}}

  59: {{7}}

  61: {{1,2,2}}

  63: {{1},{1},{1,1}}

  65: {{2},{1,2}}

  67: {{8}}

  71: {{1,1,3}}

  73: {{2,4}}

  79: {{1,5}}

  81: {{1},{1},{1},{1}}

  83: {{9}}

  87: {{1},{1,3}}

  89: {{1,1,1,2}}

  91: {{1,1},{1,2}}

  97: {{3,3}}

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&]

CROSSREFS

Cf. A001221, A048143, A056239, A112798, A286518, A286520, A290103, A302242, A303837, A304118, A304714, A304716, A305052, A305055, A305079.

Sequence in context: A248420 A011861 A229511 * A328336 A305103 A065520

Adjacent sequences:  A305075 A305076 A305077 * A305079 A305080 A305081

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 24 2018

STATUS

approved

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Last modified November 22 03:38 EST 2019. Contains 329386 sequences. (Running on oeis4.)