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A370816
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Greatest number of multisets that can be obtained by choosing a divisor of each factor in an integer factorization of n into unordered factors > 1.
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5
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1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 7, 2, 4, 4, 7, 2, 7, 2, 7, 4, 4, 2, 11, 3, 4, 5, 7, 2, 8, 2, 10, 4, 4, 4, 12, 2, 4, 4, 11, 2, 8, 2, 7, 7, 4, 2, 17, 3, 7, 4, 7, 2, 11, 4, 11, 4, 4, 2, 15, 2, 4, 7, 14, 4, 8, 2, 7, 4, 8, 2, 20, 2, 4, 7, 7, 4, 8, 2, 17, 7, 4, 2
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OFFSET
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1,2
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LINKS
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EXAMPLE
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For the factorizations of 12 we have the following choices:
(2*2*3): {{1,1,1},{1,1,2},{1,1,3},{1,2,2},{1,2,3},{2,2,3}}
(2*6): {{1,1},{1,2},{1,3},{1,6},{2,2},{2,3},{2,6}}
(3*4): {{1,1},{1,2},{1,3},{1,4},{2,3},{3,4}}
(12): {{1},{2},{3},{4},{6},{12}}
So a(12) = 7.
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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[Divisors/@#]]]&/@facs[n]], {n, 100}]
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CROSSREFS
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The version for partitions is A370808, for just prime factors A370809.
For just prime factors we have A370817.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A370813 counts non-divisor-choosable factorizations, complement A370814.
Cf. A000792, A048249, A066739, A319055, A319057, A355737, A355739, A355740, A355741, A368110, A370803.
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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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