cp's OEIS Frontend

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

%I A322794 #16 Jul 14 2019 06:30:41
%S A322794 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,
%T A322794 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,
%U A322794 2,2,1,2,1,2,2,2,2,2,1,2,3,2,1,4,2,2,2
%N A322794 Number of factorizations of n into factors > 1 where all factors have the same number of prime factors counted with multiplicity.
%C A322794 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.
%H A322794 Gus Wiseman, <a href="/A038041/a038041.txt">Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.</a>
%e A322794 The a(1260) = 13 factorizations:
%e A322794   (1260)  (18*70)   (4*9*35)   (2*2*3*3*5*7)
%e A322794           (20*63)   (6*6*35)
%e A322794           (28*45)   (4*15*21)
%e A322794           (30*42)   (6*10*21)
%e A322794           (12*105)  (6*14*15)
%e A322794                     (9*10*14)
%e A322794 The a(1260) = 13 multiset partitions:
%e A322794   {{1},{1},{2},{2},{3},{4}}
%e A322794      {{1,1},{2,2},{3,4}}
%e A322794      {{1,1},{2,3},{2,4}}
%e A322794      {{1,2},{1,2},{3,4}}
%e A322794      {{1,2},{1,3},{2,4}}
%e A322794      {{1,2},{1,4},{2,3}}
%e A322794      {{2,2},{1,3},{1,4}}
%e A322794       {{1,1,2},{2,3,4}}
%e A322794       {{1,2,2},{1,3,4}}
%e A322794       {{1,1,3},{2,2,4}}
%e A322794       {{1,1,4},{2,2,3}}
%e A322794       {{1,2,3},{1,2,4}}
%e A322794        {{1,1,2,2,3,4}}
%t A322794 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A322794 Table[Length[Select[facs[n],SameQ@@PrimeOmega/@#&]],{n,100}]
%Y A322794 Cf. A001055, A001222, A038041, A112798, A306017, A306021, A319169, A320324, A317583, A321455, A321469, A326514, A326515, A326516.
%K A322794 nonn
%O A322794 1,4
%A A322794 _Gus Wiseman_, Dec 26 2018