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A370590 Number of maximal subsets of {1..n} containing n such that it is possible to choose a different prime factor of each element (choosable).

%I A370590 #10 Feb 14 2025 05:22:02
%S A370590 0,0,1,1,1,2,3,5,2,4,14,25,13,38,46,66,28,178,57,235,106,238,656,1235,
%T A370590 288,445,2192,664,2016,6840,2300,9140,888,6236,17692,14724,7320,56000,
%U A370590 60472,70252,37160,223884,66428,290312,113172,80544,517392,1001420,114336
%N A370590 Number of maximal subsets of {1..n} containing n such that it is possible to choose a different prime factor of each element (choosable).
%C A370590 For example, the set {4,7,9,10} has choice (2,7,3,5) so is counted under a(10).
%e A370590 The a(0) = 0 through a(10) = 14 subsets (A = 10):
%e A370590   .  .  2  23  34  235  256  2357  3578  2579  237A
%e A370590                    345  356  2567  5678  4579  267A
%e A370590                         456  3457        5679  279A
%e A370590                              3567        5789  347A
%e A370590                              4567              357A
%e A370590                                                367A
%e A370590                                                378A
%e A370590                                                467A
%e A370590                                                479A
%e A370590                                                567A
%e A370590                                                579A
%e A370590                                                678A
%e A370590                                                679A
%e A370590                                                789A
%t A370590 Table[Length[Select[Subsets[Range[n],{PrimePi[n]}],MemberQ[#,n]&&Length[Select[Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#],UnsameQ@@#&]]>0&]],{n,0,10}]
%Y A370590 Not requiring n gives A370585, maximal case of A370582, complement A370583.
%Y A370590 Maximal case of A370586, complement A370587, unique A370588.
%Y A370590 An opposite version is A370591.
%Y A370590 A006530 gives greatest prime factor, least A020639.
%Y A370590 A027746 lists prime factors, indices A112798, length A001222.
%Y A370590 A355741 counts choices of a prime factor of each prime index.
%Y A370590 A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
%Y A370590 A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
%Y A370590 A368098 counts choosable unlabeled multiset partitions, complement A368097.
%Y A370590 A368100 ranks choosable multisets, complement A355529.
%Y A370590 A368414 counts choosable factorizations, complement A368413.
%Y A370590 A370592 counts choosable partitions, complement A370593.
%Y A370590 Cf. A000040, A000720, A133686, A355739, A355744, A355745, A367771, A370584, A370636, A370639.
%K A370590 nonn
%O A370590 0,6
%A A370590 _Gus Wiseman_, Feb 28 2024
%E A370590 More terms from _Jinyuan Wang_, Feb 14 2025