A327390 - OEIS

%I #6 Sep 16 2019 12:38:09

%S 1,2,2,2,2,3,2,2,3,3,2,3,2,3,3,2,2,4,2,3,4,3,2,3,3,3,4,3,2,4,2,2,3,3,

%T 3,4,2,3,4,3,2,5,2,3,4,3,2,3,3,4,3,3,2,5,3,3,4,3,2,4,2,3,6,2,4,4,2,3,

%U 3,4,2,4,2,3,4,3,3,5,2,3,5,3,2,5,3,3,4

%N Number of connected divisors of n.

%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. The maximum  connected divisor of n is A327076(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>

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

%Y See link for additional cross-references.

%Y Cf. A000005, A001221, A033273, A286518, A304714, A304716.

%K nonn

%O 1,2

%A _Gus Wiseman_, Sep 15 2019