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).

0, 0, 1, 1, 1, 2, 3, 5, 2, 4, 14, 25, 13, 38, 46, 66, 28, 178, 57, 235, 106, 238, 656, 1235, 288, 445, 2192, 664, 2016, 6840, 2300, 9140, 888, 6236, 17692, 14724, 7320, 56000, 60472, 70252, 37160, 223884, 66428, 290312, 113172, 80544, 517392, 1001420, 114336

Offset: 0

Author

Gus Wiseman, Feb 28 2024

Keywords

• nonn

Comments

For example, the set {4,7,9,10} has choice (2,7,3,5) so is counted under a(10).

Examples

			The a(0) = 0 through a(10) = 14 subsets (A = 10):
  .  .  2  23  34  235  256  2357  3578  2579  237A
                   345  356  2567  5678  4579  267A
                        456  3457        5679  279A
                             3567        5789  347A
                             4567              357A
                                               367A
                                               378A
                                               467A
                                               479A
                                               567A
                                               579A
                                               678A
                                               679A
                                               789A
		

Crossrefs

Not requiring n gives A370585, maximal case of A370582, complement A370583.

Maximal case of A370586, complement A370587, unique A370588.

An opposite version is A370591.

A006530 gives greatest prime factor, least A020639.

A027746 lists prime factors, indices A112798, length A001222.

A355741 counts choices of a prime factor of each prime index.

A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.

A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.

A368098 counts choosable unlabeled multiset partitions, complement A368097.

A368100 ranks choosable multisets, complement A355529.

A368414 counts choosable factorizations, complement A368413.

A370592 counts choosable partitions, complement A370593.

Cf. A000040, A000720, A133686, A355739, A355744, A355745, A367771, A370584, A370636, A370639.

Programs

Mathematica

Table[Length[Select[Subsets[Range[n],{PrimePi[n]}],MemberQ[#,n]&&Length[Select[Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#],UnsameQ@@#&]]>0&]],{n,0,10}]

Extensions

More terms from Jinyuan Wang, Feb 14 2025