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%I #5 Mar 11 2024 08:30:35
%S 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,
%T 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,
%U 2,3,1,3,1,2,2,2,2,3,1,2,1,2,1,4,2,2,2
%N 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.
%C First differs from A096825 at a(210) = 4, A096825(210) = 6.
%C First differs from A343943 at a(210) = 4, A343943(210) = 6.
%C First differs from A345926 at a(90) = 4, A345926(90) = 3.
%e For the factorizations of 60 we have the following choices (using prime indices {1,2,3} instead of prime factors {2,3,5}):
%e (2*2*3*5): {{1,1,2,3}}
%e (2*2*15): {{1,1,2},{1,1,3}}
%e (2*3*10): {{1,1,2},{1,2,3}}
%e (2*5*6): {{1,1,3},{1,2,3}}
%e (3*4*5): {{1,2,3}}
%e (2*30): {{1,1},{1,2},{1,3}}
%e (3*20): {{1,2},{2,3}}
%e (4*15): {{1,2},{1,3}}
%e (5*12): {{1,3},{2,3}}
%e (6*10): {{1,1},{1,2},{1,3},{2,3}}
%e (60): {{1},{2},{3}}
%e So a(60) = 4.
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t Table[Max[Length[Union[Sort/@Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#]]]&/@facs[n]],{n,100}]
%Y For all divisors (not just prime factors) we have A370816.
%Y The version for partitions is A370809, for all divisors A370808.
%Y A000005 counts divisors.
%Y A001055 counts factorizations, strict A045778.
%Y A006530 gives greatest prime factor, least A020639.
%Y A027746 lists prime factors, A112798 indices, length A001222.
%Y A355741 chooses prime factors of prime indices, variations A355744, A355745.
%Y A368413 counts non-choosable factorizations, complement A368414.
%Y A370813 counts non-divisor-choosable factorizations, complement A370814.
%Y Cf. A000792, A048249, A066739, A319055, A319057, A355529, A355535, A367771, A368100, A370585.
%K nonn
%O 1,6
%A _Gus Wiseman_, Mar 07 2024
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