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A322794
Number of factorizations of n into factors > 1 where all factors have the same number of prime factors counted with multiplicity.
18
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
1,4
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.
EXAMPLE
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}}
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 26 2018
STATUS
approved