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A370817
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Greatest number of multisets that can be obtained by choosing a prime factor of each factor in an integer factorization of n into unordered factors > 1.
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3
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1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 4, 2, 2, 2
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OFFSET
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1,6
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COMMENTS
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LINKS
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EXAMPLE
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For the factorizations of 60 we have the following choices (using prime indices {1,2,3} instead of prime factors {2,3,5}):
(2*2*3*5): {{1,1,2,3}}
(2*2*15): {{1,1,2},{1,1,3}}
(2*3*10): {{1,1,2},{1,2,3}}
(2*5*6): {{1,1,3},{1,2,3}}
(3*4*5): {{1,2,3}}
(2*30): {{1,1},{1,2},{1,3}}
(3*20): {{1,2},{2,3}}
(4*15): {{1,2},{1,3}}
(5*12): {{1,3},{2,3}}
(6*10): {{1,1},{1,2},{1,3},{2,3}}
(60): {{1},{2},{3}}
So a(60) = 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[Max[Length[Union[Sort/@Tuples[If[#==1, {}, First/@FactorInteger[#]]&/@#]]]&/@facs[n]], {n, 100}]
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CROSSREFS
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For all divisors (not just prime factors) we have A370816.
A370813 counts non-divisor-choosable factorizations, complement A370814.
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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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