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Number of subsets of {1..n} containing n such that only one set can be obtained by choosing a different prime factor of each element.
3

%I #6 Feb 29 2024 08:51:55

%S 0,0,1,2,2,6,6,18,12,20,36,104,76,284,320,408

%N Number of subsets of {1..n} containing n such that only one set can be obtained by choosing a different prime factor of each element.

%C For example, the only choice of a different prime factor of each element of (4,5,6) is (2,5,3), so {4,5,6} is counted under a(6).

%e The a(0) = 0 through a(8) = 12 subsets:

%e . . {2} {3} {4} {5} {2,6} {7} {8}

%e {2,3} {3,4} {2,5} {3,6} {2,7} {3,8}

%e {3,5} {4,6} {3,7} {5,8}

%e {4,5} {2,5,6} {4,7} {6,8}

%e {2,3,5} {3,5,6} {5,7} {7,8}

%e {3,4,5} {4,5,6} {2,3,7} {3,5,8}

%e {2,5,7} {3,7,8}

%e {2,6,7} {5,6,8}

%e {3,4,7} {5,7,8}

%e {3,5,7} {6,7,8}

%e {3,6,7} {3,5,7,8}

%e {4,5,7} {5,6,7,8}

%e {4,6,7}

%e {2,3,5,7}

%e {2,5,6,7}

%e {3,4,5,7}

%e {3,5,6,7}

%e {4,5,6,7}

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

%Y First differences of A370584, cf. A370582, complement A370583.

%Y For any number of choices we have A370586, complement A370587.

%Y For binary indices see A370638, A370639, complement A370589.

%Y A006530 gives greatest prime factor, least A020639.

%Y A027746 lists prime factors, indices A112798, length A001222.

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

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

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

%Y A368098 counts choosable unlabeled multiset partitions, complement A368097.

%Y A368100 ranks choosable multisets, complement A355529.

%Y A368414 counts choosable factorizations, complement A368413.

%Y A370585 counts maximal choosable sets.

%Y A370592 counts choosable partitions, complement A370593.

%Y A370636 counts choosable subsets for binary indices, complement A370637.

%Y Cf. A000040, A000720, A005117, A045778, A133686, A355739, A355744, A355745, A367771, A367905.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Feb 28 2024