A326083 - OEIS

%I #16 Mar 24 2022 17:34:50

%S 1,2,3,5,7,12,16,27,37,58,80,131,171,277,380,580,785,1250,1655,2616,

%T 3516,5344,7257,11353,14931,23204,31379,47511,63778,98681,130503,

%U 201357,270038,407429,548090,840171,1110429,1701872,2284325,3440337,4601656

%N Number of subsets of {1..n} containing all of their pairwise sums <= n.

%C 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).

%C From _Gus Wiseman_, Jul 08 2019: (Start)

%C 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:

%C   {}  {}   {}   {}     {}     {}       {}

%C       {1}  {1}  {1}    {1}    {1}      {1}

%C            {2}  {2}    {2}    {2}      {2}

%C                 {3}    {3}    {3}      {3}

%C                 {2,3}  {4}    {4}      {4}

%C                        {2,3}  {5}      {5}

%C                        {3,4}  {2,3}    {6}

%C                               {2,5}    {2,3}

%C                               {3,4}    {2,5}

%C                               {3,5}    {3,4}

%C                               {4,5}    {3,5}

%C                               {3,4,5}  {4,5}

%C                                        {4,6}

%C                                        {5,6}

%C                                        {3,4,5}

%C                                        {4,5,6}

%C (End)

%H Fausto A. C. Cariboni, <a href="/A326083/b326083.txt">Table of n, a(n) for n = 0..100</a>

%F For n > 0, a(n) = A103580(n) + 1.

%e The a(0) = 1 through a(6) = 16 subsets:

%e   {}  {}   {}     {}       {}         {}           {}

%e       {1}  {2}    {2}      {3}        {3}          {4}

%e            {1,2}  {3}      {4}        {4}          {5}

%e                   {2,3}    {2,4}      {5}          {6}

%e                   {1,2,3}  {3,4}      {2,4}        {3,6}

%e                            {2,3,4}    {3,4}        {4,5}

%e                            {1,2,3,4}  {3,5}        {4,6}

%e                                       {4,5}        {5,6}

%e                                       {2,4,5}      {2,4,6}

%e                                       {3,4,5}      {3,4,6}

%e                                       {2,3,4,5}    {3,5,6}

%e                                       {1,2,3,4,5}  {4,5,6}

%e                                                    {2,4,5,6}

%e                                                    {3,4,5,6}

%e                                                    {2,3,4,5,6}

%e                                                    {1,2,3,4,5,6}

%e The a(7) = 27 subsets:

%e   {}  {4}  {36}  {246}  {2467}  {24567}  {234567}  {1234567}

%e       {5}  {45}  {356}  {3467}  {34567}

%e       {6}  {46}  {367}  {3567}

%e       {7}  {47}  {456}  {4567}

%e            {56}  {457}

%e            {57}  {467}

%e            {67}  {567}

%t Table[Length[Select[Subsets[Range[n]],SubsetQ[#,Select[Plus@@@Tuples[#,2],#<=n&]]&]],{n,0,10}]

%Y Cf. A007865, A050291, A051026, A054519, A085489, A093971, A103580, A120641, A151897, A326020, A326023, A326076, A326080.

%K nonn

%O 0,2

%A _Gus Wiseman_, Jun 05 2019