A326083 Number of subsets of {1..n} containing all of their pairwise sums <= n.

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

Offset: 0

Author

Gus Wiseman, Jun 05 2019

Keywords

• nonn

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)

Examples

			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}
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..100

Crossrefs

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

Programs

Mathematica

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

Formula

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