A327524 Number of factorizations of the n-th uniform number A072774(n) into uniform numbers > 1.

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 1, 2, 2, 5, 1, 1, 2, 2, 1, 2, 2, 3, 1, 5, 1, 7, 2, 2, 2, 7, 1, 2, 2, 1, 5, 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 1, 2, 11, 2, 5, 1, 2, 5, 1, 1, 2, 2, 5, 1, 5, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 7, 1, 5, 1, 5, 2, 1, 1, 5, 2, 1, 5, 2, 2, 2

Offset: 1

Author

Gus Wiseman, Sep 17 2019

Keywords

• nonn

Comments

A number is uniform if its prime multiplicities are all equal, meaning it is a power of a squarefree number. Uniform numbers are listed in A072774.

Examples

			The a(31) = 7 factorizations of 36 into uniform numbers together with the corresponding multiset partitions of {1,1,2,2}:
  (2*2*3*3)  {{1},{1},{2},{2}}
  (2*2*9)    {{1},{1},{2,2}}
  (2*3*6)    {{1},{2},{1,2}}
  (3*3*4)    {{2},{2},{1,1}}
  (4*9)      {{1,1},{2,2}}
  (6*6)      {{1,2},{1,2}}
  (36)       {{1,1,2,2}}
		

Links

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

Crossrefs

See link for additional cross-references.

Cf. A000961, A001055, A005117, A007947, A071625, A112798.

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