A327390 Number of connected divisors of n.

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, 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, 3, 4, 2, 4, 2, 3, 4, 3, 3, 5, 2, 3, 5, 3, 2, 5, 3, 3, 4

Offset: 1

Author

Gus Wiseman, Sep 15 2019

Keywords

• nonn

Comments

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).

Links

• Gus Wiseman, Sequences counting and encoding certain classes of multisets

Crossrefs

See link for additional cross-references.

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

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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]]]]]]]]];
Table[Length[Select[Divisors[n],Length[zsm[primeMS[#]]]<=1&]],{n,100}]