A326498 -id:A326498 - cp's OEIS frontend

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

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

nonn

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

			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)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..80
P. J. Cameron and P. Erdős, On the number of integers with various properties, in R. A. Mullin, ed., Number Theory: Proc. First Conf. of Canad. Number Theory Assoc. Conf., Banff, De Gruyter, Berlin, 1990, pp. 61-79.
N. Hindman and H. Jordan, Measures of sum-free intersecting families, New York J. Math. 13 (2007), 97-106.

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.

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 *)

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

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

A326497 Number of maximal sum-free and product-free subsets of {1..n}.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 6, 8, 9, 15, 21, 26, 38, 51, 69, 89, 119, 149, 197, 261, 356, 447, 601, 781, 1003, 1293, 1714, 2228, 2931, 3697, 4843, 6258, 8187, 10273, 13445, 16894, 21953, 27469, 35842, 45410, 58948, 73939, 95199, 120593, 154510, 192995, 247966, 312642

Offset: 0

Gus Wiseman, Jul 09 2019

nonn

A set is sum-free and product-free if it contains no sum or product of two (not necessarily distinct) elements.

			The a(2) = 1 through a(10) = 15 subsets (A = 10):
  {2}  {23}  {23}  {23}   {23}   {237}   {256}   {267}    {23A}
             {34}  {25}   {256}  {256}   {258}   {345}    {345}
                   {345}  {345}  {267}   {267}   {357}    {34A}
                          {456}  {345}   {345}   {2378}   {357}
                                 {357}   {357}   {2569}   {38A}
                                 {4567}  {2378}  {2589}   {2378}
                                         {4567}  {4567}   {2569}
                                         {5678}  {4679}   {2589}
                                                 {56789}  {267A}
                                                          {269A}
                                                          {4567}
                                                          {4679}
                                                          {479A}
                                                          {56789}
                                                          {6789A}

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..68
Andrew Howroyd, PARI Program

Sum-free and product-free subsets are A326495.
Sum-free subsets are A007865.
Maximal sum-free subsets are A121269.
Product-free subsets are A326489.
Maximal product-free subsets are A326496.
Subsets with sums (and products) are A326083.
Cf. A051026, A103580, A325710, A326076, A326117, A326491, A326492, A326498.

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

\\ See link for program file.
for(n=0, 37, print1(A326497(n), ", ")) \\ Andrew Howroyd, Aug 30 2019

a(21)-a(40) from Andrew Howroyd, Aug 30 2019

a(41)-a(48) from Jinyuan Wang, Oct 11 2020

A325710 Number of maximal subsets of {1..n} containing no products of distinct elements.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 4, 4, 6, 6, 10, 10, 14, 14, 24, 28, 32, 32, 62, 62, 92, 102, 184, 184, 254, 254, 474, 506, 686, 686, 1172, 1172, 1792, 1906, 3568, 3794, 5326, 5326, 10282, 10618, 14822, 14822, 25564, 25564, 35304, 39432, 76888, 76888, 100574, 100574, 197870, 201622, 282014

Offset: 0

Gus Wiseman, Jul 09 2019

nonn

			The a(1) = 1 through a(9) = 6 maximal subsets:
  {1}  {1}  {1}   {1}    {1}     {1}     {1}      {1}       {1}
       {2}  {23}  {234}  {2345}  {2345}  {23457}  {23457}   {234579}
                                 {2456}  {24567}  {23578}   {235789}
                                 {3456}  {34567}  {24567}   {245679}
                                                  {25678}   {256789}
                                                  {345678}  {3456789}

Andrew Howroyd, PARI Program

Subsets without products of distinct elements are A326117.
Maximal product-free subsets are A326496.
Subsets with products are A326076.
Maximal subsets without sums of distinct elements are A326498.
Maximal subsets without quotients are A326492.
Maximal subsets without sums or products of distinct elements are A326025.
Cf. A121269, A151897, A326116, A326489, A326497, A326024.

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],Intersection[#,Times@@@Subsets[#,{2,n}]]=={}&]]],{n,0,10}]

\\ See link for program file.
for(n=0, 30, print1(A325710(n), ", ")) \\ Andrew Howroyd, Aug 29 2019

Terms a(16) and beyond from Andrew Howroyd, Aug 29 2019

A326025 Number of maximal subsets of {1..n} containing no sums or products of distinct elements.

Original entry on oeis.org

1, 1, 2, 2, 2, 4, 5, 10, 13, 20, 28, 40, 54, 82, 120, 172, 244, 347, 471, 651, 874, 1198, 1635, 2210, 2867, 3895, 5234, 6889, 9019, 11919, 15629, 20460, 26254, 33827, 43881, 56367, 71841, 91834, 117695, 148503, 188039, 311442, 390859, 488327, 610685, 759665

Offset: 0

Gus Wiseman, Jul 09 2019

nonn

			The a(1) = 1 through a(8) = 13 maximal subsets:
  {1}  {1}  {1}    {1}      {1}      {1}        {1}        {1}
       {2}  {2,3}  {2,3,4}  {2,3,4}  {2,3,4}    {2,3,4}    {2,3,4}
                            {2,4,5}  {2,4,5}    {2,3,7}    {2,4,5}
                            {3,4,5}  {2,5,6}    {2,4,5}    {2,4,7}
                                     {3,4,5,6}  {2,4,7}    {2,5,6}
                                                {2,5,6}    {2,5,8}
                                                {2,6,7}    {2,6,7}
                                                {3,4,5,6}  {2,3,7,8}
                                                {3,5,6,7}  {3,4,5,6}
                                                {4,5,6,7}  {3,4,6,8}
                                                           {3,5,6,7}
                                                           {3,6,7,8}
                                                           {4,5,6,7,8}

Andrew Howroyd, PARI Program

Maximal subsets without sums of distinct elements are A326498.
Maximal subsets without products of distinct elements are A325710.
Subsets without sums or products of distinct elements are A326024.
Subsets with sums (and products) are A326083.
Maximal sum-free and product-free subsets are A326497.
Cf. A007865, A051026, A121269, A151897, A326116, A326117, A326491, A326495, A326496.

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

\\ See link for program file.
for(n=0, 25, print1(A326025(n), ", ")) \\ Andrew Howroyd, Aug 29 2019

a(16)-a(40) from Andrew Howroyd, Aug 29 2019

a(41)-a(45) from Jinyuan Wang, Oct 03 2020

A326024 Number of subsets of {1..n} containing no sums or products of distinct elements.

Original entry on oeis.org

1, 2, 3, 5, 9, 15, 25, 41, 68, 109, 179, 284, 443, 681, 1062, 1587, 2440, 3638, 5443, 8021, 11953, 17273, 25578, 37001, 53953, 77429, 113063, 160636, 232928, 330775, 475380, 672056, 967831, 1359743, 1952235, 2743363, 3918401, 5495993, 7856134, 10984547, 15669741

Offset: 0

Gus Wiseman, Jul 09 2019

nonn

			The a(0) = 1 through a(5) = 15 subsets:
  {}  {}   {}   {}     {}       {}
      {1}  {1}  {1}    {1}      {1}
           {2}  {2}    {2}      {2}
                {3}    {3}      {3}
                {2,3}  {4}      {4}
                       {2,3}    {5}
                       {2,4}    {2,3}
                       {3,4}    {2,4}
                       {2,3,4}  {2,5}
                                {3,4}
                                {3,5}
                                {4,5}
                                {2,3,4}
                                {2,4,5}
                                {3,4,5}

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

Subsets without sums of distinct elements are A151897.
Subsets without products of distinct elements are A326117.
Maximal subsets without sums or products of distinct elements are A326025.
Subsets with sums (and products) are A326083.
Sum-free and product-free subsets are A326495.
Cf. A007865, A051026, A121269, A325710, A326076, A326489, A326497, A326498.

Table[Length[Select[Subsets[Range[n]],Intersection[#,Union[Plus@@@Subsets[#,{2,n}],Times@@@Subsets[#,{2,n}]]]=={}&]],{n,0,10}]

a(n)={
   my(recurse(k, es, ep)=
    if(k > n, 1,
      my(t = self()(k + 1, es, ep));
      if(!bittest(es,k) && !bittest(ep,k),
         es = bitor(es, bitand((2<Andrew Howroyd, Aug 25 2019

Terms a(16)-a(40) from Andrew Howroyd, Aug 25 2019

cp's OEIS Frontend

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

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A326497 Number of maximal sum-free and product-free subsets of {1..n}.

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A325710 Number of maximal subsets of {1..n} containing no products of distinct elements.

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A326025 Number of maximal subsets of {1..n} containing no sums or products of distinct elements.

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A326024 Number of subsets of {1..n} containing no sums or products of distinct elements.

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