A327398 - OEIS

%I #5 Oct 20 2019 21:44:51

%S 1,2,3,2,5,3,7,2,3,5,11,3,13,7,5,2,17,3,19,5,21,11,23,3,5,13,3,7,29,5,

%T 31,2,11,17,7,3,37,19,39,5,41,21,43,11,5,23,47,3,7,5,17,13,53,3,11,7,

%U 57,29,59,5,61,31,21,2,65,11,67,17,23,7,71,3,73,37

%N Maximum connected squarefree divisor of n.

%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>

%e The connected squarefree divisors of 189 are {1, 3, 7, 21}, so a(189) = 21.

%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 Table[Max[Select[Divisors[n],SquareFreeQ[#]&&Length[zsm[primeMS[#]]]<=1&]],{n,100}]

%Y The maximum connected divisor of n is A327076(n).

%Y The maximum squarefree divisor of n is A007947(n).

%Y Connected numbers are A305078.

%Y Connected squarefree numbers are A328513.

%Y Cf. A000005, A001221, A302242, A302494, A304714, A305079, A327656, A328514.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 20 2019