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

1,5

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.

Links

Table of n, a(n) for n=1..87.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

Example

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}}

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}]

Crossrefs

See link for additional cross-references.

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

Sequence in context: A242442 A163768 A327659 * A287917 A325615 A029434

Adjacent sequences: A327516 A327517 A327518 * A327520 A327521 A327522

Keyword

nonn

Author

Gus Wiseman, Sep 21 2019

Status

approved