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A322794
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Number of factorizations of n into factors > 1 where all factors have the same number of prime factors counted with multiplicity.
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15
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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
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OFFSET
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1,4
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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}}
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MATHEMATICA
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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}]
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CROSSREFS
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Cf. A001055, A001222, A038041, A112798, A306017, A306021, A319169, A320324, A317583, A321455, A321469, A326514, A326515, A326516.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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