A322794 Number of factorizations of n into factors > 1 where all factors have the same number of prime factors counted with multiplicity.

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 4, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 2, 4, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 3, 2, 1, 4, 2, 2, 2

Offset: 1

Author

Gus Wiseman, Dec 26 2018

Keywords

• nonn

Comments

Also the number of uniform multiset partitions of the multiset of prime indices of n, where a multiset partition is uniform if all parts have the same size.

Examples

			The a(1260) = 13 factorizations:
  (1260)  (18*70)   (4*9*35)   (2*2*3*3*5*7)
          (20*63)   (6*6*35)
          (28*45)   (4*15*21)
          (30*42)   (6*10*21)
          (12*105)  (6*14*15)
                    (9*10*14)
The a(1260) = 13 multiset partitions:
  {{1},{1},{2},{2},{3},{4}}
     {{1,1},{2,2},{3,4}}
     {{1,1},{2,3},{2,4}}
     {{1,2},{1,2},{3,4}}
     {{1,2},{1,3},{2,4}}
     {{1,2},{1,4},{2,3}}
     {{2,2},{1,3},{1,4}}
      {{1,1,2},{2,3,4}}
      {{1,2,2},{1,3,4}}
      {{1,1,3},{2,2,4}}
      {{1,1,4},{2,2,3}}
      {{1,2,3},{1,2,4}}
       {{1,1,2,2,3,4}}
		

Links

• Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Crossrefs

Cf. A001055, A001222, A038041, A112798, A306017, A306021, A319169, A320324, A317583, A321455, A321469, A326514, A326515, A326516.

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],SameQ@@PrimeOmega/@#&]],{n,100}]