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 A326083 Number of subsets of {1..n} containing all of their pairwise sums <= n. 16
 1, 2, 3, 5, 7, 12, 16, 27, 37, 58, 80, 131, 171, 277, 380, 580, 785, 1250, 1655, 2616, 3516, 5344, 7257, 11353, 14931, 23204, 31379, 47511, 63778, 98681, 130503, 201357, 270038, 407429, 548090, 840171, 1110429, 1701872, 2284325, 3440337, 4601656 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The summands are allowed to be equal. The case where they must be distinct is A326080. If A007865 counts sum-free sets, this sequence counts sum-closed sets. This is different from sum-full sets (A093971). From Gus Wiseman, Jul 08 2019: (Start) Also the number of subsets of {1..n} containing no sum of any multiset of the elements. For example, the a(0) = 1 through a(6) = 16 subsets are:   {}  {}   {}   {}     {}     {}       {}       {1}  {1}  {1}    {1}    {1}      {1}            {2}  {2}    {2}    {2}      {2}                 {3}    {3}    {3}      {3}                 {2,3}  {4}    {4}      {4}                        {2,3}  {5}      {5}                        {3,4}  {2,3}    {6}                               {2,5}    {2,3}                               {3,4}    {2,5}                               {3,5}    {3,4}                               {4,5}    {3,5}                               {3,4,5}  {4,5}                                        {4,6}                                        {5,6}                                        {3,4,5}                                        {4,5,6} (End) LINKS FORMULA For n > 0, a(n) = A103580(n) + 1. EXAMPLE The a(0) = 1 through a(6) = 16 subsets:   {}  {}   {}     {}       {}         {}           {}       {1}  {2}    {2}      {3}        {3}          {4}            {1,2}  {3}      {4}        {4}          {5}                   {2,3}    {2,4}      {5}          {6}                   {1,2,3}  {3,4}      {2,4}        {3,6}                            {2,3,4}    {3,4}        {4,5}                            {1,2,3,4}  {3,5}        {4,6}                                       {4,5}        {5,6}                                       {2,4,5}      {2,4,6}                                       {3,4,5}      {3,4,6}                                       {2,3,4,5}    {3,5,6}                                       {1,2,3,4,5}  {4,5,6}                                                    {2,4,5,6}                                                    {3,4,5,6}                                                    {2,3,4,5,6}                                                    {1,2,3,4,5,6} The a(7) = 27 subsets:   {}  {4}  {36}  {246}  {2467}  {24567}  {234567}  {1234567}       {5}  {45}  {356}  {3467}  {34567}       {6}  {46}  {367}  {3567}       {7}  {47}  {456}  {4567}            {56}  {457}            {57}  {467}            {67}  {567} MATHEMATICA Table[Length[Select[Subsets[Range[n]], SubsetQ[#, Select[Plus@@@Tuples[#, 2], #<=n&]]&]], {n, 0, 10}] CROSSREFS Cf. A007865, A050291, A051026, A054519, A085489, A093971, A103580, A120641, A151897, A326020, A326023, A326076, A326080. Sequence in context: A319635 A179822 A319769 * A027959 A060730 A308928 Adjacent sequences:  A326080 A326081 A326082 * A326084 A326085 A326086 KEYWORD nonn AUTHOR Gus Wiseman, Jun 05 2019 STATUS approved

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Last modified December 12 22:06 EST 2019. Contains 329963 sequences. (Running on oeis4.)