A327518 Number of factorizations of A302696(n), the n-th number that is 1, 2, or a nonprime number with pairwise coprime prime indices, into factors > 1 satisfying the same conditions.

1, 1, 2, 1, 3, 1, 2, 1, 1, 5, 2, 1, 4, 1, 2, 2, 7, 1, 1, 1, 1, 4, 2, 1, 7, 1, 2, 1, 4, 1, 5, 1, 11, 2, 2, 1, 2, 1, 2, 1, 7, 1, 1, 1, 4, 2, 1, 1, 1, 12, 2, 4, 1, 2, 7, 2, 1, 1, 10, 1, 1, 2, 15, 5, 1, 4, 2, 5, 1, 1, 1, 1, 1, 2, 4, 2, 1, 1, 12, 1, 2, 1, 1, 2, 2

Offset: 1

Author

Gus Wiseman, Sep 19 2019

Keywords

• nonn

Examples

			The a(59) = 10 factorizations of 120 using the allowed factors, together with the corresponding multiset partitions of {1,1,1,2,3}:
  (2*2*2*15)  {{1},{1},{1},{2,3}}
  (2*2*30)    {{1},{1},{1,2,3}}
  (2*4*15)    {{1},{1,1},{2,3}}
  (2*6*10)    {{1},{1,2},{1,3}}
  (2*60)      {{1},{1,1,2,3}}
  (4*30)      {{1,1},{1,2,3}}
  (6*20)      {{1,2},{1,1,3}}
  (8*15)      {{1,1,1},{2,3}}
  (10*12)     {{1,3},{1,1,2}}
  (120)       {{1,1,1,2,3}}
		

Links

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

Crossrefs

See link for additional cross-references.

Cf. A001055, A056239, A112798, A302569, A302696, A304711, A305079, A327392.

Programs

Mathematica

nn=100;
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,#]&]}]];
y=Select[Range[nn],#==1||CoprimeQ@@primeMS[#]&];
Table[Length[facsusing[Rest[y],n]],{n,y}]