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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

Table of n, a(n) for n=0..40.

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.)