cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A326496 Number of maximal product-free subsets of {1..n}.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 6, 6, 9, 9, 15, 17, 30, 30, 46, 46, 51, 61, 103, 103, 129, 158, 282, 282, 322, 322, 553, 553, 615, 689, 1247, 1365, 1870, 1870, 3566, 3758, 5244, 5244, 8677, 8677, 9807, 12147, 23351, 23351, 27469, 31694, 45718, 47186, 54594, 54594, 95382, 108198
Offset: 0

Views

Author

Gus Wiseman, Jul 09 2019

Keywords

Comments

A set is product-free if it contains no product of two (not necessarily distinct) elements.
Also the number of maximal quotient-free subsets of {1..n}.

Examples

			The a(2) = 1 through a(10) = 6 subsets (A = 10):
  {2}  {23}  {23}  {235}  {235}   {2357}   {23578}   {23578}   {23578}
             {34}  {345}  {256}   {2567}   {25678}   {256789}  {2378A}
                          {3456}  {34567}  {345678}  {345678}  {256789}
                                                     {456789}  {26789A}
                                                               {345678A}
                                                               {456789A}

Links

Crossrefs

Product-free subsets are A326489.
Subsets without products of distinct elements are A326117.
Maximal sum-free subsets are A121269.
Maximal sum-free and product-free subsets are A326497.
Maximal subsets without products of distinct elements are A325710.
Cf. A007865, A051026, A326076, A326491, A326492, A326495, A327591.

Programs

  • Mathematica
    fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],Intersection[#,Times@@@Tuples[#,2]]=={}&]]],{n,0,10}]
  • PARI
    \\ See link for program file.
for(n=0, 30, print1(A326496(n), ", ")) \\ Andrew Howroyd, Aug 30 2019

Extensions

a(18)-a(55) from Andrew Howroyd, Aug 30 2019

A326495 Number of subsets of {1..n} containing no sums or products of pairs of elements.

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 17, 30, 45, 71, 101, 171, 258, 427, 606, 988, 1328, 2141, 3116, 4952, 6955, 11031, 15320, 23978, 33379, 48698, 66848, 104852, 144711, 220757, 304132, 461579, 636555, 973842, 1316512, 1958827, 2585432, 3882842, 5237092, 7884276, 10555738, 15729292
Offset: 0

Views

Author

Gus Wiseman, Jul 09 2019

Keywords

Comments

The pairs are not required to be strict.

Examples

			The a(1) = 1 through a(6) = 17 subsets:
  {}  {}   {}     {}     {}       {}
      {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}    {2,6}
                         {4,5}    {3,4}
                         {3,4,5}  {3,5}
                                  {4,5}
                                  {4,6}
                                  {5,6}
                                  {2,5,6}
                                  {3,4,5}
                                  {4,5,6}

Links

Crossrefs

Subsets without sums are A007865.
Subsets without products are A326489.
Subsets without differences or quotients are A326490.
Maximal subsets without sums or products are A326497.
Subsets with sums (and products) are A326083.
Cf. A051026, A103580, A326020, A326076, A326117, A326491.

Programs

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

Formula

For n > 0, a(n) = A326490(n) - 1.

Extensions

a(19)-a(41) from Andrew Howroyd, Aug 25 2019

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

Views

Author

Gus Wiseman, Jul 09 2019

Keywords

Comments

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

Examples

			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}

Links

Crossrefs

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.

Programs

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

Extensions

a(21)-a(40) from Andrew Howroyd, Aug 30 2019
a(41)-a(48) from Jinyuan Wang, Oct 11 2020

A326490 Number of subsets of {1..n} containing no differences or quotients of pairs of distinct elements.

Original entry on oeis.org

1, 2, 3, 5, 7, 12, 18, 31, 46, 72, 102, 172, 259, 428, 607, 989, 1329, 2142, 3117, 4953, 6956, 11032, 15321, 23979, 33380, 48699, 66849, 104853, 144712, 220758, 304133, 461580, 636556, 973843, 1316513, 1958828, 2585433, 3882843, 5237093, 7884277, 10555739, 15729293
Offset: 0

Views

Author

Gus Wiseman, Jul 09 2019

Keywords

Examples

			The a(0) = 1 through a(6) = 18 subsets:
  {}  {}   {}   {}     {}     {}       {}
      {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}    {2,6}
                              {4,5}    {3,4}
                              {3,4,5}  {3,5}
                                       {4,5}
                                       {4,6}
                                       {5,6}
                                       {2,5,6}
                                       {3,4,5}
                                       {4,5,6}

Links

Crossrefs

Subsets without difference are A007865.
Maximal subsets without differences or quotients are A326491.
Subsets without quotients are A327591.
Subsets with differences and quotients are A326494.
Cf. A051026, A054519, A325860, A326023, A326079, A326489.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],Intersection[#,Union[Divide@@@Reverse/@Subsets[#,{2}],Subtract@@@Reverse/@Subsets[#,{2}]]]=={}&]],{n,0,10}]
  • PARI
    a(n)={
   my(recurse(k, b)=
    if(k > n, 1,
      my(t = self()(k + 1, b));
      for(i=1, k\2, if(bittest(b,i) && (bittest(b,k-i) || (!(k%i) && bittest(b,k/i))), return(t)));
      t += self()(k + 1, b + (1<Andrew Howroyd, Aug 25 2019

Formula

For n > 0, a(n) = A326495(n) + 1.

Extensions

a(19)-a(41) from Andrew Howroyd, Aug 25 2019

A327591 Number of subsets of {1..n} containing no quotients of pairs of distinct elements.

Original entry on oeis.org

1, 2, 3, 5, 7, 13, 23, 45, 89, 137, 253, 505, 897, 1793, 3393, 6353, 9721, 19441, 35665, 71329, 129953, 247233, 477665, 955329, 1700417, 2657281, 5184001, 10368001, 19407361, 38814721, 68868353, 137736705, 260693505, 505830401, 999641601, 1882820609, 2807196673
Offset: 0

Views

Author

Peter Kagey, Sep 17 2019

Keywords

Examples

			The a(0) = 1 through a(5) = 13 subsets:
  {}  {}   {}   {}     {}     {}
      {1}  {1}  {1}    {1}    {1}
           {2}  {2}    {2}    {2}
                {3}    {3}    {3}
                {2,3}  {4}    {4}
                       {2,3}  {5}
                       {3,4}  {2,3}
                              {2,5}
                              {3,4}
                              {3,5}
                              {4,5}
                              {2,3,5}
                              {3,4,5}

Links

Crossrefs

Maximal subsets without quotients are A326492.
Subsets with quotients are A326023.
Subsets without differences or quotients are A326490.
Subsets without products are A326489.
Cf. A007865, A051026, A325860, A325994, A326079, A326117, A326491, A326496.

Formula

A326489(n) + 1 for n > 0.

A326492 Number of maximal subsets of {1..n} containing no quotients of pairs of distinct elements.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 4, 5, 7, 7, 10, 10, 16, 18, 31, 31, 47, 47, 52, 62, 104, 104, 130, 159, 283, 283, 323, 323, 554, 554, 616, 690, 1248, 1366, 1871, 1871, 3567, 3759, 5245, 5245, 8678, 8678, 9808, 12148, 23352, 23352, 27470, 31695, 45719, 47187, 54595, 54595, 95383, 108199
Offset: 0

Views

Author

Gus Wiseman, Jul 09 2019

Keywords

Examples

			The a(0) = 1 through a(9) = 5 subsets:
  {}  {1}  {1}  {1}   {1}   {1}    {1}     {1}      {1}       {1}
           {2}  {23}  {23}  {235}  {235}   {2357}   {23578}   {23578}
                      {34}  {345}  {256}   {2567}   {25678}   {256789}
                                   {3456}  {34567}  {345678}  {345678}
                                                              {456789}

Crossrefs

Subsets with quotients are A326023.
Subsets with quotients > 1 are A326079.
Subsets without quotients are A327591.
Maximal subsets without differences or quotients are A326491.
Maximal subsets without quotients (or products) are A326496.
Cf. A007865, A121269, A325860, A325994, A326117, A326489, A326490.

Programs

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

Formula

a(n) = A326496(n) + 1 for n > 1. - Andrew Howroyd, Aug 30 2019

Extensions

Terms a(16) and beyond from Andrew Howroyd, Aug 30 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

Views

Author

Gus Wiseman, Jul 09 2019

Keywords

Examples

			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}

Links

Crossrefs

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.

Programs

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

Extensions

a(16)-a(40) from Andrew Howroyd, Aug 29 2019
a(41)-a(45) from Jinyuan Wang, Oct 03 2020

A326494 Number of subsets of {1..n} containing all differences and quotients of pairs of distinct elements.

Original entry on oeis.org

1, 2, 4, 6, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127
Offset: 0

Views

Author

Gus Wiseman, Jul 09 2019

Keywords

Comments

The only allowed sets are the empty set, any singleton, any initial interval of positive integers and {2,4}. This can be shown by induction. - Andrew Howroyd, Aug 25 2019

Examples

			The a(0) = 1 through a(6) = 13 subsets:
  {}  {}   {}     {}       {}         {}           {}
      {1}  {1}    {1}      {1}        {1}          {1}
           {2}    {2}      {2}        {2}          {2}
           {1,2}  {3}      {3}        {3}          {3}
                  {1,2}    {4}        {4}          {4}
                  {1,2,3}  {1,2}      {5}          {5}
                           {2,4}      {1,2}        {6}
                           {1,2,3}    {2,4}        {1,2}
                           {1,2,3,4}  {1,2,3}      {2,4}
                                      {1,2,3,4}    {1,2,3}
                                      {1,2,3,4,5}  {1,2,3,4}
                                                   {1,2,3,4,5}
                                                   {1,2,3,4,5,6}

Crossrefs

Subsets with difference are A054519.
Subsets with quotients are A326023.
Subsets with quotients > 1 are A326079.
Subsets without differences or quotients are A326490.
Cf. A005408, A007865, A051026, A325849, A326076, A326491.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],SubsetQ[#,Union[Divide@@@Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&],Subtract@@@Select[Tuples[#,2],Greater@@#&]]]&]],{n,0,10}]

Formula

a(n) = 2*n + 1 = A005408(n) for n > 3. - Andrew Howroyd, Aug 25 2019

Extensions

Terms a(20) and beyond from Andrew Howroyd, Aug 25 2019
Showing 1-8 of 8 results.

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