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Connected squarefree numbers.
5

%I #4 Oct 22 2019 10:20:08

%S 1,2,3,5,7,11,13,17,19,21,23,29,31,37,39,41,43,47,53,57,59,61,65,67,

%T 71,73,79,83,87,89,91,97,101,103,107,109,111,113,115,127,129,131,133,

%U 137,139,149,151,157,159,163,167,173,179,181,183,185,191,193,195

%N Connected squarefree numbers.

%C First differs from A318718 and A318719 in having 195 = prime(2) * prime(3) * prime(6).

%C A squarefree number with prime factorization prime(m_1) * ... * prime(m_k) is connected if the simple labeled graph with vertex set {m_1,...,m_k} and edges between any two vertices with a common divisor greater than 1 is connected. Connected numbers are listed in A305078.

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vSX9dPMGJhxB8rOknCGvOs6PiyhupdWNpqLsnphdgU6MEVqFBnWugAXidDhwHeKqZe_YnUqYeGOXsOk/pub">Sequences counting and encoding certain classes of multisets</a>

%F Intersection of A005117 and A305078.

%e The sequence of all connected sets of multisets together with their MM-numbers (A302242) begins:

%e 1: {}

%e 2: {{}}

%e 3: {{1}}

%e 5: {{2}}

%e 7: {{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 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 53: {{1,1,1,1}}

%e 57: {{1},{1,1,1}}

%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[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];

%t Select[Range[100],SquareFreeQ[#]&&Length[zsm[primeMS[#]]]<=1&]

%Y A subset of A005117.

%Y These are Heinz numbers of the partitions counted by A304714.

%Y The maximum connected squarefree divisor of n is A327398(n).

%Y Cf. A056239, A112798, A286518, A302242, A304716, A305078, A305079, A327076, A328514.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 20 2019