cp's OEIS Frontend

A328513 Connected squarefree numbers.

%I A328513 #4 Oct 22 2019 10:20:08
%S A328513 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 A328513 71,73,79,83,87,89,91,97,101,103,107,109,111,113,115,127,129,131,133,
%U A328513 137,139,149,151,157,159,163,167,173,179,181,183,185,191,193,195
%N A328513 Connected squarefree numbers.
%C A328513 First differs from A318718 and A318719 in having 195 = prime(2) * prime(3) * prime(6).
%C A328513 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 A328513 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 A328513 Intersection of A005117 and A305078.
%e A328513 The sequence of all connected sets of multisets together with their MM-numbers (A302242) begins:
%e A328513    1: {}
%e A328513    2: {{}}
%e A328513    3: {{1}}
%e A328513    5: {{2}}
%e A328513    7: {{1,1}}
%e A328513   11: {{3}}
%e A328513   13: {{1,2}}
%e A328513   17: {{4}}
%e A328513   19: {{1,1,1}}
%e A328513   21: {{1},{1,1}}
%e A328513   23: {{2,2}}
%e A328513   29: {{1,3}}
%e A328513   31: {{5}}
%e A328513   37: {{1,1,2}}
%e A328513   39: {{1},{1,2}}
%e A328513   41: {{6}}
%e A328513   43: {{1,4}}
%e A328513   47: {{2,3}}
%e A328513   53: {{1,1,1,1}}
%e A328513   57: {{1},{1,1,1}}
%t A328513 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A328513 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 A328513 Select[Range[100],SquareFreeQ[#]&&Length[zsm[primeMS[#]]]<=1&]
%Y A328513 A subset of A005117.
%Y A328513 These are Heinz numbers of the partitions counted by A304714.
%Y A328513 The maximum connected squarefree divisor of n is A327398(n).
%Y A328513 Cf. A056239, A112798, A286518, A302242, A304716, A305078, A305079, A327076, A328514.
%K A328513 nonn
%O A328513 1,2
%A A328513 _Gus Wiseman_, Oct 20 2019