cp's OEIS Frontend

A370586 Number of subsets of {1..n} containing n such that it is possible to choose a different prime factor of each element (choosable).

%I A370586 #11 Feb 27 2024 15:09:29
%S A370586 0,0,1,2,2,6,8,20,12,20,44,116,88,320,380,508,264,1792,968,4552,3136,
%T A370586 5600,10056,27896,11792,16384,46688,19584,48288,198528,110928,507984,
%U A370586 99648,463552,859376,821136,470688,3730368,4033920,4651296,2932512,19078464
%N A370586 Number of subsets of {1..n} containing n such that it is possible to choose a different prime factor of each element (choosable).
%e A370586 The a(0) = 0 through a(7) = 20 subsets:
%e A370586   .  .  {2}  {3}    {4}    {5}      {6}      {7}
%e A370586              {2,3}  {3,4}  {2,5}    {2,6}    {2,7}
%e A370586                            {3,5}    {3,6}    {3,7}
%e A370586                            {4,5}    {4,6}    {4,7}
%e A370586                            {2,3,5}  {5,6}    {5,7}
%e A370586                            {3,4,5}  {2,5,6}  {6,7}
%e A370586                                     {3,5,6}  {2,3,7}
%e A370586                                     {4,5,6}  {2,5,7}
%e A370586                                              {2,6,7}
%e A370586                                              {3,4,7}
%e A370586                                              {3,5,7}
%e A370586                                              {3,6,7}
%e A370586                                              {4,5,7}
%e A370586                                              {4,6,7}
%e A370586                                              {5,6,7}
%e A370586                                              {2,3,5,7}
%e A370586                                              {2,5,6,7}
%e A370586                                              {3,4,5,7}
%e A370586                                              {3,5,6,7}
%e A370586                                              {4,5,6,7}
%t A370586 Table[Length[Select[Subsets[Range[n]], MemberQ[#,n]&&Length[Select[Tuples[If[#==1, {},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]>0&]],{n,0,10}]
%Y A370586 First differences of A370582, complement A370583, cf. A370584.
%Y A370586 Maximal choosable sets are counted by A370585.
%Y A370586 The complement is counted by A370587.
%Y A370586 For a unique choice we have A370588.
%Y A370586 For binary indices instead of prime factors we have A370639.
%Y A370586 A006530 gives greatest prime factor, least A020639.
%Y A370586 A027746 lists prime factors, indices A112798, length A001222.
%Y A370586 A355741 counts choices of a prime factor of each prime index.
%Y A370586 A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
%Y A370586 A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
%Y A370586 A368098 counts choosable unlabeled multiset partitions, complement A368097.
%Y A370586 A368100 ranks choosable multisets, complement A355529.
%Y A370586 A368414 counts choosable factorizations, complement A368413.
%Y A370586 A370592 counts choosable partitions, complement A370593.
%Y A370586 Cf. A000040, A000720, A005117, A045778, A133686, A355739, A355744, A355745, A367771, A367905, A370636.
%K A370586 nonn
%O A370586 0,4
%A A370586 _Gus Wiseman_, Feb 26 2024
%E A370586 a(19)-a(41) from _Alois P. Heinz_, Feb 27 2024