A327519 Number of factorizations of A305078(n - 1), the n-th connected number, into connected numbers > 1.

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 4, 2, 1, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 4, 2, 3, 1, 2, 1, 2, 1, 1, 4, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 7, 1, 1, 4, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 7, 2, 1

Offset: 1

Author

Gus Wiseman, Sep 21 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.

Examples

			The a(190) = 8 factorizations of 585 together with the corresponding multiset partitions of {2,2,3,6}:
  (3*3*5*13)  {{2},{2},{3},{6}}
  (3*3*65)    {{2},{2},{3,6}}
  (3*5*39)    {{2},{3},{2,6}}
  (3*195)     {{2},{2,3,6}}
  (5*9*13)    {{3},{2,2},{6}}
  (5*117)     {{3},{2,2,6}}
  (9*65)      {{2,2},{3,6}}
  (585)       {{2,2,3,6}}
		

Links

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

Crossrefs

See link for additional cross-references.

Cf. A286518, A302569, A304714, A304716, A305078, A305079, A327076.

Programs

Mathematica

nn=100;
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]]]]]]]]];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
y=Select[Range[nn],Length[zsm[primeMS[#]]]==1&];
Table[Length[facsusing[y,n]],{n,y}]