A327523 Number of factorizations of the n-th number with distinct prime multiplicities A130091(n) into numbers > 1 with distinct prime multiplicities.

1, 1, 1, 2, 1, 1, 3, 2, 1, 3, 1, 5, 1, 3, 1, 3, 1, 5, 2, 3, 3, 1, 1, 7, 1, 5, 1, 1, 3, 3, 1, 9, 2, 3, 3, 1, 5, 5, 1, 1, 3, 11, 1, 3, 1, 11, 1, 3, 3, 1, 9, 5, 1, 5, 1, 3, 14, 1, 3, 3, 1, 1, 5, 1, 11, 1, 9, 1, 3, 3, 2, 3, 3, 1, 15, 1, 5, 5, 1, 1, 20, 3, 3, 1, 1

Offset: 1

Author

Gus Wiseman, Sep 16 2019

Keywords

• nonn

Comments

A number's prime multiplicities are also called its (unsorted) prime signature.

Examples

			The a(57) = 14 factorizations of 96 together with the corresponding multiset partitions of {1,1,1,1,1,2}:
  (2*2*2*2*2*3)  {{1}{1}{1}{1}{1}{2}}
  (2*2*2*3*4)    {{1}{1}{1}{2}{11}}
  (2*2*2*12)     {{1}{1}{1}{112}}
  (2*2*3*8)      {{1}{1}{2}{111}}
  (2*2*24)       {{1}{1}{1112}}
  (2*3*4*4)      {{1}{2}{11}{11}}
  (2*3*16)       {{1}{2}{1111}}
  (2*4*12)       {{1}{11}{112}}
  (2*48)         {{1}{11112}}
  (3*4*8)        {{2}{11}{111}}
  (3*32)         {{2}{11111}}
  (4*24)         {{11}{1112}}
  (8*12)         {{111}{112}}
  (96)           {{111112}}
		

Links

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

Crossrefs

See link for additional cross-references.

Cf. A098859, A112798, A130091, A255231.

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],UnsameQ@@Last/@FactorInteger[#]&];
Table[Length[facsusing[Rest[y],n]],{n,y}]