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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.

%I A370588 #10 Mar 28 2025 14:01:06
%S A370588 0,0,1,2,2,6,6,18,12,20,36,104,76,284,320,408,252,1548,872,3968,2800,
%T A370588 4704,8568,24008,10832,14832,40688,18240,43632,176240,97344,449824,
%U A370588 95328,404992,760752,698864,436464,3296048,3564576,4057904,2677776,16892352,8676576
%N 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.
%C A370588 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 A370588 The a(0) = 0 through a(8) = 12 subsets:
%e A370588   .  .  {2}  {3}    {4}    {5}      {2,6}    {7}        {8}
%e A370588              {2,3}  {3,4}  {2,5}    {3,6}    {2,7}      {3,8}
%e A370588                            {3,5}    {4,6}    {3,7}      {5,8}
%e A370588                            {4,5}    {2,5,6}  {4,7}      {6,8}
%e A370588                            {2,3,5}  {3,5,6}  {5,7}      {7,8}
%e A370588                            {3,4,5}  {4,5,6}  {2,3,7}    {3,5,8}
%e A370588                                              {2,5,7}    {3,7,8}
%e A370588                                              {2,6,7}    {5,6,8}
%e A370588                                              {3,4,7}    {5,7,8}
%e A370588                                              {3,5,7}    {6,7,8}
%e A370588                                              {3,6,7}    {3,5,7,8}
%e A370588                                              {4,5,7}    {5,6,7,8}
%e A370588                                              {4,6,7}
%e A370588                                              {2,3,5,7}
%e A370588                                              {2,5,6,7}
%e A370588                                              {3,4,5,7}
%e A370588                                              {3,5,6,7}
%e A370588                                              {4,5,6,7}
%t A370588 Table[Length[Select[Subsets[Range[n]],MemberQ[#,n] && Length[Select[Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]==1&]],{n,0,10}]
%Y A370588 First differences of A370584, cf. A370582, complement A370583.
%Y A370588 For any number of choices we have A370586, complement A370587.
%Y A370588 For binary indices see A370638, A370639, complement A370589.
%Y A370588 A006530 gives greatest prime factor, least A020639.
%Y A370588 A027746 lists prime factors, indices A112798, length A001222.
%Y A370588 A355741 counts choices of a prime factor of each prime index.
%Y A370588 A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
%Y A370588 A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
%Y A370588 A368098 counts choosable unlabeled multiset partitions, complement A368097.
%Y A370588 A368100 ranks choosable multisets, complement A355529.
%Y A370588 A368414 counts choosable factorizations, complement A368413.
%Y A370588 A370585 counts maximal choosable sets.
%Y A370588 A370592 counts choosable partitions, complement A370593.
%Y A370588 A370636 counts choosable subsets for binary indices, complement A370637.
%Y A370588 Cf. A000040, A000720, A005117, A045778, A133686, A355739, A355744, A355745, A367771, A367905.
%K A370588 nonn
%O A370588 0,4
%A A370588 _Gus Wiseman_, Feb 28 2024
%E A370588 More terms from _Jinyuan Wang_, Mar 28 2025