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.

Previous Showing 11-19 of 19 results.

A326116 Number of subsets of {2..n} containing no products of two or more distinct elements.

Original entry on oeis.org

1, 2, 4, 8, 16, 28, 56, 100, 200, 364, 728, 1232, 2464, 4592, 8296, 15920, 31840, 55952, 111904, 195712, 362336, 697360, 1394720, 2334112, 4668224, 9095392, 17225312, 31242784, 62485568, 106668608, 213337216, 392606528, 755131840, 1491146912, 2727555424, 4947175712
Offset: 1

Views

Author

Gus Wiseman, Jun 06 2019

Keywords

Comments

First differs from A308542 at a(12) = 1232, A308542(12) = 1184.
If this sequence counts product-free sets, A308542 counts product-closed sets.

Examples

			The a(6) = 28 subsets:
  {}  {2}  {2,3}  {2,3,4}  {2,3,4,5}
      {3}  {2,4}  {2,3,5}  {2,4,5,6}
      {4}  {2,5}  {2,4,5}  {3,4,5,6}
      {5}  {2,6}  {2,4,6}
      {6}  {3,4}  {2,5,6}
           {3,5}  {3,4,5}
           {3,6}  {3,4,6}
           {4,5}  {3,5,6}
           {4,6}  {4,5,6}
           {5,6}

Links

Crossrefs

Cf. A007865, A051026, A103580, A196724, A326020, A326023, A326076, A326078, A326079, A326081, A326117, A308542.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[2,n]],Intersection[#,Select[Times@@@Subsets[#,{2}],#<=n&]]=={}&]],{n,10}]
  • PARI
    a(n)={
   my(recurse(k, ep)=
    if(k > n, 1,
      my(t = self()(k + 1, ep));
      if(!bittest(ep,k),
         forstep(i=n\k, 1, -1, if(bittest(ep,i), ep=bitor(ep,1<<(k*i))));
         t += self()(k + 1, ep);
      );
      t);
   );
   recurse(2, 2);
} \\ Andrew Howroyd, Aug 25 2019

Formula

For n > 0, a(n) = A326117(n) - 1.

Extensions

Terms a(21)-a(36) from Andrew Howroyd, Aug 25 2019

A326491 Number of maximal subsets of {1..n} containing no differences or quotients of pairs of distinct elements.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 7, 9, 10, 16, 22, 27, 39, 52, 70, 90, 120, 150, 198, 262, 357, 448, 602, 782, 1004, 1294, 1715, 2229, 2932, 3698, 4844, 6259, 8188, 10274, 13446, 16895, 21954, 27470, 35843, 45411, 58949, 73940, 95200, 120594, 154511, 192996, 247967, 312643
Offset: 0

Views

Author

Gus Wiseman, Jul 09 2019

Keywords

Examples

			The a(1) = 1 through a(9) = 10 subsets:
  {1}  {1}  {1}    {1}    {1}      {1}      {1}        {1}        {1}
       {2}  {2,3}  {2,3}  {2,3}    {2,3}    {2,3,7}    {2,5,6}    {2,6,7}
                   {3,4}  {2,5}    {2,5,6}  {2,5,6}    {2,5,8}    {3,4,5}
                          {3,4,5}  {3,4,5}  {2,6,7}    {2,6,7}    {3,5,7}
                                   {4,5,6}  {3,4,5}    {3,4,5}    {2,3,7,8}
                                            {3,5,7}    {3,5,7}    {2,5,6,9}
                                            {4,5,6,7}  {2,3,7,8}  {2,5,8,9}
                                                       {4,5,6,7}  {4,5,6,7}
                                                       {5,6,7,8}  {4,6,7,9}
                                                                  {5,6,7,8,9}

Crossrefs

Subsets without differences or quotients are A326490.
Subsets with differences and quotients are A326494.
Maximal subsets without differences are A121269
Maximal subsets without quotients are A326492.
Cf. A007865, A051026, A054519, A326023, A326117, A326489, A326496, A327591.

Programs

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

Formula

a(n) = A326497(n) + 1 for n > 1. - Andrew Howroyd, Aug 30 2019

Extensions

a(16)-a(40) from Andrew Howroyd, Aug 30 2019
a(41)-a(48) from Jinyuan Wang, Mar 04 2025

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

A326114 Number of subsets of {2..n} containing no product of two or more (not necessarily distinct) elements.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 22, 44, 76, 116, 222, 444, 788, 1576, 3068, 5740, 8556, 17112, 31752, 63504, 116176, 221104, 438472, 876944, 1569424, 2447664, 4869576, 9070920, 17022360, 34044720, 61923312, 123846624, 234698720, 462007072, 922838192, 1734564112, 2591355792, 5182711584
Offset: 0

Views

Author

Gus Wiseman, Jun 06 2019

Keywords

Comments

The strict case is A326117.
Also the number of subsets of {2..n} containing all of their integer products <= n. For example, the a(1) = 1 through a(5) = 12 subsets are:
{} {} {} {} {} {}
{2} {2} {3} {3}
{3} {4} {4}
{2,3} {2,4} {5}
{3,4} {2,4}
{2,3,4} {3,4}
{3,5}
{4,5}
{2,3,4}
{2,4,5}
{3,4,5}
{2,3,4,5}

Examples

			The a(1) = 1 through a(5) = 12 subsets:
  {}  {}   {}     {}     {}
      {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}

Crossrefs

Cf. A007865, A051026, A103580, A196724, A326020, A326023, A326076, A326078, A326079, A326081, A326116, A326117.

Formula

a(n > 0) = A326076(n)/2.

Extensions

Terms a(21) and beyond from Andrew Howroyd, Aug 30 2019

A308542 Number of subsets of {2..n} containing the product of any set of distinct elements whose product is <= n.

Original entry on oeis.org

1, 2, 4, 8, 16, 28, 56, 100, 200, 364, 728, 1184, 2368, 4448, 8056, 15008, 30016, 52736, 105472, 183424, 339840, 663616, 1327232, 2217088, 4434176, 8744320, 16559168, 30034624, 60069248, 103402112, 206804224, 379941440, 730800064, 1454649248, 2659869664, 4786282208
Offset: 1

Views

Author

Gus Wiseman, Jun 06 2019

Keywords

Comments

First differs from A326116 at a(12) = 1184, A326116(12) = 1232.
If this sequence counts product-closed sets, A326116 counts product-free sets.

Examples

			The a(6) = 28 sets:
  {}  {2}  {2,4}  {2,3,6}  {2,3,4,6}  {2,3,4,5,6}
      {3}  {2,5}  {2,4,5}  {2,3,5,6}
      {4}  {2,6}  {2,4,6}  {2,4,5,6}
      {5}  {3,4}  {2,5,6}  {3,4,5,6}
      {6}  {3,5}  {3,4,5}
           {3,6}  {3,4,6}
           {4,5}  {3,5,6}
           {4,6}  {4,5,6}
           {5,6}

Crossrefs

Cf. A007865, A051026, A103580, A196724, A326020, A326023, A326076, A326078, A326079, A326081, A326116, A326117.

Programs

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

Formula

For n > 0, a(n) = A326081(n)/2.

Extensions

Terms a(21) and beyond from Andrew Howroyd, Aug 24 2019

A326082 Number of maximal sets of pairwise indivisible divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 5, 2, 3, 3, 5, 2, 5, 2, 5, 3, 3, 2, 8, 3, 3, 4, 5, 2, 7, 2, 6, 3, 3, 3, 9, 2, 3, 3, 8, 2, 7, 2, 5, 5, 3, 2, 12, 3, 5, 3, 5, 2, 8, 3, 8, 3, 3, 2, 15, 2, 3, 5, 7, 3, 7, 2, 5, 3, 7, 2, 15, 2, 3, 5, 5, 3, 7, 2, 12, 5, 3, 2, 15, 3
Offset: 1

Views

Author

Gus Wiseman, Jun 05 2019

Keywords

Comments

Depends only on prime signature.
The non-maximal case is A096827.

Examples

			The maximal sets of pairwise indivisible divisors of n = 1, 2, 4, 8, 12, 24, 30, 32, 36, 48, 60 are:
   1   1   1   1   1     1      1         1    1       1       1
       2   2   2   12    24     30        2    36      48      60
           4   4   2,3   2,3    5,6       4    2,3     2,3     2,15
               8   3,4   3,4    2,15      8    2,9     3,4     3,20
                   4,6   3,8    3,10      16   3,4     3,8     4,30
                         4,6    2,3,5     32   4,18    4,6     5,12
                         6,8    6,10,15        9,12    6,8     2,3,5
                         8,12                  12,18   3,16    3,4,5
                                               4,6,9   6,16    4,5,6
                                                       8,12    3,4,10
                                                       12,16   6,15,20
                                                       16,24   10,12,15
                                                               12,15,20
                                                               12,20,30
                                                               4,6,10,15

Crossrefs

Cf. A001055, A051026, A067992, A096827, A143824, A285572, A285573, A303362, A305148, A305149, A316476, A325861, A326023, A326077.

Programs

  • Mathematica
    stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Rest[Subsets[Divisors[n]]],stableQ[#,Divisible]&]]],{n,100}]

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
Previous Showing 11-19 of 19 results.

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