A327519 - OEIS

%I #5 Sep 22 2019 08:05:47

%S 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,

%T 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,

%U 1,1,4,1,1,2,1,2,1,1,1,1,2,2,1,1,7,2,1

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

%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.

%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>

%e The a(190) = 8 factorizations of 585 together with the corresponding multiset partitions of {2,2,3,6}:

%e   (3*3*5*13)  {{2},{2},{3},{6}}

%e   (3*3*65)    {{2},{2},{3,6}}

%e   (3*5*39)    {{2},{3},{2,6}}

%e   (3*195)     {{2},{2,3,6}}

%e   (5*9*13)    {{3},{2,2},{6}}

%e   (5*117)     {{3},{2,2,6}}

%e   (9*65)      {{2,2},{3,6}}

%e   (585)       {{2,2,3,6}}

%t nn=100;

%t 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]]]]]]]]];

%t 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,#]&]}]];

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t y=Select[Range[nn],Length[zsm[primeMS[#]]]==1&];

%t Table[Length[facsusing[y,n]],{n,y}]

%Y See link for additional cross-references.

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

%K nonn

%O 1,5

%A _Gus Wiseman_, Sep 21 2019