A033273 -id:A033273 - cp's OEIS frontend

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

Gus Wiseman, Sep 15 2019

nonn

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).

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

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

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}]

A327391 Number of divisors of n that are 1, prime, or whose prime indices are pairwise coprime.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 4, 5, 2, 4, 2, 6, 3, 4, 2, 8, 2, 4, 2, 6, 2, 8, 2, 6, 4, 4, 4, 6, 2, 4, 3, 8, 2, 6, 2, 6, 4, 4, 2, 10, 2, 4, 4, 6, 2, 4, 4, 8, 3, 4, 2, 12, 2, 4, 3, 7, 3, 8, 2, 6, 4, 8, 2, 8, 2, 4, 4, 6, 4, 6, 2, 10, 2, 4, 2, 9, 4, 4

Offset: 1

Gus Wiseman, Sep 20 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. Numbers that are prime or whose prime indices are pairwise coprime are listed in A302569.

			The divisors of 84 that are 1, prime, or whose prime indices are pairwise coprime are {1, 2, 3, 4, 6, 7, 12, 14, 28}, so a(84) = 9.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A000005, A006530, A033273, A056239, A112798, A302569, A304711, A327513.

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

cp's OEIS Frontend

A327390 Number of connected divisors of n.

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