A370588 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.

0, 0, 1, 2, 2, 6, 6, 18, 12, 20, 36, 104, 76, 284, 320, 408, 252, 1548, 872, 3968, 2800, 4704, 8568, 24008, 10832, 14832, 40688, 18240, 43632, 176240, 97344, 449824, 95328, 404992, 760752, 698864, 436464, 3296048, 3564576, 4057904, 2677776, 16892352, 8676576

Offset: 0

Author

Gus Wiseman, Feb 28 2024

Keywords

• nonn

Comments

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).

Examples

			The a(0) = 0 through a(8) = 12 subsets:
  .  .  {2}  {3}    {4}    {5}      {2,6}    {7}        {8}
             {2,3}  {3,4}  {2,5}    {3,6}    {2,7}      {3,8}
                           {3,5}    {4,6}    {3,7}      {5,8}
                           {4,5}    {2,5,6}  {4,7}      {6,8}
                           {2,3,5}  {3,5,6}  {5,7}      {7,8}
                           {3,4,5}  {4,5,6}  {2,3,7}    {3,5,8}
                                             {2,5,7}    {3,7,8}
                                             {2,6,7}    {5,6,8}
                                             {3,4,7}    {5,7,8}
                                             {3,5,7}    {6,7,8}
                                             {3,6,7}    {3,5,7,8}
                                             {4,5,7}    {5,6,7,8}
                                             {4,6,7}
                                             {2,3,5,7}
                                             {2,5,6,7}
                                             {3,4,5,7}
                                             {3,5,6,7}
                                             {4,5,6,7}
		

Crossrefs

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

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

For binary indices see A370638, A370639, complement A370589.

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.

A370585 counts maximal choosable sets.

A370592 counts choosable partitions, complement A370593.

A370636 counts choosable subsets for binary indices, complement A370637.

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

Programs

Mathematica

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

Extensions

More terms from Jinyuan Wang, Mar 28 2025