A370586 - OEIS

A370586

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

0, 0, 1, 2, 2, 6, 8, 20, 12, 20, 44, 116, 88, 320, 380, 508, 264, 1792, 968, 4552, 3136, 5600, 10056, 27896, 11792, 16384, 46688, 19584, 48288, 198528, 110928, 507984, 99648, 463552, 859376, 821136, 470688, 3730368, 4033920, 4651296, 2932512, 19078464

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OFFSET

0,4

LINKS

Table of n, a(n) for n=0..41.

EXAMPLE

The a(0) = 0 through a(7) = 20 subsets:

. . {2} {3} {4} {5} {6} {7}

{2,3} {3,4} {2,5} {2,6} {2,7}

{3,5} {3,6} {3,7}

{4,5} {4,6} {4,7}

{2,3,5} {5,6} {5,7}

{3,4,5} {2,5,6} {6,7}

{3,5,6} {2,3,7}

{4,5,6} {2,5,7}

{2,6,7}

{3,4,7}

{3,5,7}

{3,6,7}

{4,5,7}

{4,6,7}

{5,6,7}

{2,3,5,7}

{2,5,6,7}

{3,4,5,7}

{3,5,6,7}

{4,5,6,7}

MATHEMATICA

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

CROSSREFS

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

Maximal choosable sets are counted by A370585.

The complement is counted by A370587.

For a unique choice we have A370588.

For binary indices instead of prime factors we have A370639.

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, A005117, A045778, A133686, A355739, A355744, A355745, A367771, A367905, A370636.

Sequence in context: A054153 A000673 A355640 * A276425 A129383 A052957

Adjacent sequences: A370583 A370584 A370585 * A370587 A370588 A370589

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 26 2024

EXTENSIONS

a(19)-a(41) from Alois P. Heinz, Feb 27 2024

STATUS

approved