cp's OEIS Frontend

A370817 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.

%I A370817 #8 Sep 17 2024 13:03:09
%S A370817 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 A370817 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 A370817 2,3,1,3,1,2,2,2,2,3,1,2,1,2,1,4,2,2,2
%N A370817 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 A370817 First differs from A096825 at a(210) = 4, A096825(210) = 6.
%C A370817 First differs from A343943 at a(210) = 4, A343943(210) = 6.
%C A370817 First differs from A345926 at a(90) = 4, A345926(90) = 3.
%H A370817 Max Alekseyev, <a href="/A370817/b370817.txt">Table of n, a(n) for n = 1..10000</a>
%e A370817 For the factorizations of 60 we have the following choices (using prime indices {1,2,3} instead of prime factors {2,3,5}):
%e A370817   (2*2*3*5): {{1,1,2,3}}
%e A370817    (2*2*15): {{1,1,2},{1,1,3}}
%e A370817    (2*3*10): {{1,1,2},{1,2,3}}
%e A370817     (2*5*6): {{1,1,3},{1,2,3}}
%e A370817     (3*4*5): {{1,2,3}}
%e A370817      (2*30): {{1,1},{1,2},{1,3}}
%e A370817      (3*20): {{1,2},{2,3}}
%e A370817      (4*15): {{1,2},{1,3}}
%e A370817      (5*12): {{1,3},{2,3}}
%e A370817      (6*10): {{1,1},{1,2},{1,3},{2,3}}
%e A370817        (60): {{1},{2},{3}}
%e A370817 So a(60) = 4.
%t A370817 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A370817 Table[Max[Length[Union[Sort/@Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#]]]&/@facs[n]],{n,100}]
%Y A370817 For all divisors (not just prime factors) we have A370816.
%Y A370817 The version for partitions is A370809, for all divisors A370808.
%Y A370817 A000005 counts divisors.
%Y A370817 A001055 counts factorizations, strict A045778.
%Y A370817 A006530 gives greatest prime factor, least A020639.
%Y A370817 A027746 lists prime factors, A112798 indices, length A001222.
%Y A370817 A355741 chooses prime factors of prime indices, variations A355744, A355745.
%Y A370817 A368413 counts non-choosable factorizations, complement A368414.
%Y A370817 A370813 counts non-divisor-choosable factorizations, complement A370814.
%Y A370817 Cf. A000792, A048249, A066739, A319055, A319057, A355529, A355535, A367771, A368100, A370585.
%K A370817 nonn
%O A370817 1,6
%A A370817 _Gus Wiseman_, Mar 07 2024