A121269 Number of maximal sum-free subsets of {1,2,...,n}.

1, 1, 2, 2, 4, 5, 6, 8, 13, 17, 23, 29, 37, 51, 66, 86, 118, 158, 201, 265, 359, 471, 598, 797, 1043, 1378, 1765, 2311, 3064, 3970, 5017, 6537, 8547, 11020, 14007, 18026, 23404, 30026, 37989, 48945, 62759, 80256, 101070, 129193, 164835, 209279, 262693, 334127

Offset

0,3

Comments

Also the number of maximal subsets of {1..n} containing no differences of pairs of elements. - Gus Wiseman, Jul 10 2019

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..80

N. Hindman and H. Jordan, Measures of sum-free intersecting families, New York J. Math. 13 (2007), 97-106.

Example

a(5)=5 because the maximal sum-free subsets of {1,2,3,4,5} are {1,4}, {2,3}, {2,5}, {1,3,5} and {3,4,5}
From Gus Wiseman, Jul 10 2019: (Start)
The a(1) = 1 through a(8) = 13 subsets:
  {1}  {1}  {1,3}  {1,3}  {1,4}    {2,3}    {1,4,6}    {1,3,8}
       {2}  {2,3}  {1,4}  {2,3}    {1,3,5}  {1,4,7}    {1,4,6}
                   {2,3}  {2,5}    {1,4,6}  {2,3,7}    {1,4,7}
                   {3,4}  {1,3,5}  {2,5,6}  {2,5,6}    {1,5,8}
                          {3,4,5}  {3,4,5}  {2,6,7}    {1,6,8}
                                   {4,5,6}  {3,4,5}    {2,5,6}
                                            {1,3,5,7}  {2,5,8}
                                            {4,5,6,7}  {2,6,7}
                                                       {3,4,5}
                                                       {1,3,5,7}
                                                       {2,3,7,8}
                                                       {4,5,6,7}
                                                       {5,6,7,8}
(End)

Mathematica

fasmax[y_]:=Complement[y, Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]], Intersection[#, Plus@@@Tuples[#, 2]]=={}&]]], {n, 0, 10}] (* Gus Wiseman, Jul 10 2019 *)

Crossrefs

Maximal product-free subsets are A326496.

Sum-free subsets are A007865.

Maximal sum-free and product-free subsets are A326497.

Subsets with sums are A326083.

Maximal subsets without sums of distinct elements are A326498.

Cf. A103580, A326020, A326489, A326495.

Sequence in context: A238687 A238433 A238424 * A211860 A250114 A056219

Adjacent sequences: A121266 A121267 A121268 * A121270 A121271 A121272

Keyword

nonn

Author

N. Hindman (nhindman(AT)aol.com), Aug 23 2006

Extensions

a(0) = 1 prepended by Gus Wiseman, Jul 10 2019

Terms a(42) and beyond from Fausto A. C. Cariboni, Oct 26 2020

Status

approved