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%I #10 Sep 26 2019 02:30:58
%S 1,2,3,2,5,3,7,2,9,5,11,3,13,7,5,2,17,9,19,5,21,11,23,3,25,13,27,7,29,
%T 5,31,2,11,17,7,9,37,19,39,5,41,21,43,11,9,23,47,3,49,25,17,13,53,27,
%U 11,7,57,29,59,5,61,31,63,2,65,11,67,17,23,7,71,9,73
%N Maximum divisor of n that is 1 or connected.
%C A number n with prime factorization n = prime(m_1)^s_1 * ... * prime(m_k)^s_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, which is the union of this sequence without 1.
%C Also the maximum MM-number (A302242) of a connected subset of the multiset of multisets with MM-number n.
%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 If n is in A305078, then a(n) = n.
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t zsm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],GCD@@s[[#]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
%t Table[Max[Select[Divisors[n],Length[zsm[primeMS[#]]]<=1&]],{n,30}]
%Y Positions of prime numbers are A302569.
%Y Connected numbers are A305078.
%Y Cf. A007947, A056239, A112798, A286518, A302242, A304716, A305079, A322338, A322390, A322391.
%K nonn
%O 1,2
%A _Gus Wiseman_, Sep 05 2019
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