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%I #10 Aug 03 2021 01:55:00
%S 2,3,5,7,9,11,13,17,19,21,23,25,27,29,31,37,39,41,43,47,49,53,57,59,
%T 61,63,65,67,71,73,79,81,83,87,89,91,97,101,103,107,109,111,113,115,
%U 117,121,125,127,129,131,133,137,139,147,149,151,157,159,163,167
%N Heinz numbers of connected integer partitions.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C 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.
%H Madeline Locus Dawsey, Tyler Russell and Dannie Urban, <a href="https://arxiv.org/abs/2108.00943">Polynomials Associated to Integer Partitions</a>, arXiv:2108.00943 [math.NT], 2021.
%e The sequence of all connected multiset multisystems (see A302242, A112798) begins:
%e 2: {{}}
%e 3: {{1}}
%e 5: {{2}}
%e 7: {{1,1}}
%e 9: {{1},{1}}
%e 11: {{3}}
%e 13: {{1,2}}
%e 17: {{4}}
%e 19: {{1,1,1}}
%e 21: {{1},{1,1}}
%e 23: {{2,2}}
%e 25: {{2},{2}}
%e 27: {{1},{1},{1}}
%e 29: {{1,3}}
%e 31: {{5}}
%e 37: {{1,1,2}}
%e 39: {{1},{1,2}}
%e 41: {{6}}
%e 43: {{1,4}}
%e 47: {{2,3}}
%e 49: {{1,1},{1,1}}
%e 53: {{1,1,1,1}}
%e 57: {{1},{1,1,1}}
%e 59: {{7}}
%e 61: {{1,2,2}}
%e 63: {{1},{1},{1,1}}
%e 65: {{2},{1,2}}
%e 67: {{8}}
%e 71: {{1,1,3}}
%e 73: {{2,4}}
%e 79: {{1,5}}
%e 81: {{1},{1},{1},{1}}
%e 83: {{9}}
%e 87: {{1},{1,3}}
%e 89: {{1,1,1,2}}
%e 91: {{1,1},{1,2}}
%e 97: {{3,3}}
%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 Select[Range[300],Length[zsm[primeMS[#]]]==1&]
%Y Cf. A001221, A048143, A056239, A112798, A286518, A286520, A290103, A302242, A303837, A304118, A304714, A304716, A305052, A305055, A305079.
%K nonn
%O 1,1
%A _Gus Wiseman_, May 24 2018
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