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 1-7 of 7 results.

A326117 Number of subsets of {1..n} containing no products of two or more distinct elements.

Original entry on oeis.org

1, 2, 3, 5, 9, 17, 29, 57, 101, 201, 365, 729, 1233, 2465, 4593, 8297, 15921, 31841, 55953, 111905, 195713, 362337, 697361, 1394721, 2334113, 4668225, 9095393, 17225313, 31242785, 62485569, 106668609, 213337217, 392606529, 755131841, 1491146913, 2727555425, 4947175713
Offset: 0

Views

Author

Gus Wiseman, Jun 06 2019

Keywords

Comments

If this sequence counts product-free sets, A326081 counts product-closed sets.

Examples

			The a(6) = 28 sets:
  {}  {1}  {2,3}  {2,3,4}  {2,3,4,5}
      {2}  {2,4}  {2,3,5}  {2,4,5,6}
      {3}  {2,5}  {2,4,5}  {3,4,5,6}
      {4}  {2,6}  {2,4,6}
      {5}  {3,4}  {2,5,6}
      {6}  {3,5}  {3,4,5}
           {3,6}  {3,4,6}
           {4,5}  {3,5,6}
           {4,6}  {4,5,6}
           {5,6}

Links

Crossrefs

Cf. A007865, A051026, A103580, A196724, A326020, A326023, A326076, A326078, A326079, A326081, A326116, A308542.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],Intersection[#,Select[Times@@@Subsets[#,{2}],#<=n&]]=={}&]],{n,0,20}]

Formula

For n > 0, a(n) = A326116(n) + 1.

Extensions

Terms a(21)-a(36) from Andrew Howroyd, Aug 30 2019

A326489 Number of product-free subsets of {1..n}.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 22, 44, 88, 136, 252, 504, 896, 1792, 3392, 6352, 9720, 19440, 35664, 71328, 129952, 247232, 477664, 955328, 1700416, 2657280, 5184000, 10368000, 19407360, 38814720, 68868352, 137736704, 260693504, 505830400, 999641600, 1882820608, 2807196672
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.

Examples

			The a(0) = 1 through a(6) = 22 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}
                             {2,3,5}  {3,5}
                             {3,4,5}  {3,6}
                                      {4,5}
                                      {4,6}
                                      {5,6}
                                      {2,3,5}
                                      {2,5,6}
                                      {3,4,5}
                                      {3,4,6}
                                      {3,5,6}
                                      {4,5,6}
                                      {3,4,5,6}

Links

Crossrefs

Product-closed subsets are A326076.
Subsets containing no products are A326114.
Subsets containing no products of distinct elements are A326117.
Subsets containing no quotients are A327591.
Maximal product-free subsets are A326496.
Cf. A007865, A051026, A326023, A326081, A326116, A326495.

Programs

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

Extensions

a(21)-a(36) from Andrew Howroyd, Aug 25 2019
a(0)=1 prepended to data, example and b-file by Peter Kagey, Sep 18 2019

A326081 Number of subsets of {1..n} containing the product of any set of distinct elements whose product is <= n.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 56, 112, 200, 400, 728, 1456, 2368, 4736, 8896, 16112, 30016, 60032, 105472, 210944, 366848, 679680, 1327232, 2654464, 4434176, 8868352, 17488640, 33118336, 60069248, 120138496, 206804224, 413608448, 759882880, 1461600128, 2909298496, 5319739328
Offset: 0

Views

Author

Gus Wiseman, Jun 05 2019

Keywords

Comments

For n > 0, this sequence divided by 2 first differs from A326116 at a(12)/2 = 1184, A326116(12) = 1232.
If A326117 counts product-free sets, this sequence counts product-closed sets.
The non-strict case is A326076.

Examples

			The a(6) = 56 subsets:
  {}  {1}  {1,2}  {1,2,4}  {1,2,3,6}  {1,2,3,4,6}  {1,2,3,4,5,6}
      {2}  {1,3}  {1,2,5}  {1,2,4,5}  {1,2,3,5,6}
      {3}  {1,4}  {1,2,6}  {1,2,4,6}  {1,2,4,5,6}
      {4}  {1,5}  {1,3,4}  {1,2,5,6}  {1,3,4,5,6}
      {5}  {1,6}  {1,3,5}  {1,3,4,5}  {2,3,4,5,6}
      {6}  {2,4}  {1,3,6}  {1,3,4,6}
           {2,5}  {1,4,5}  {1,3,5,6}
           {2,6}  {1,4,6}  {1,4,5,6}
           {3,4}  {1,5,6}  {2,3,4,6}
           {3,5}  {2,3,6}  {2,3,5,6}
           {3,6}  {2,4,5}  {2,4,5,6}
           {4,5}  {2,4,6}  {3,4,5,6}
           {4,6}  {2,5,6}
           {5,6}  {3,4,5}
                  {3,4,6}
                  {3,5,6}
                  {4,5,6}

Crossrefs

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

Programs

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

Formula

For n > 0, a(n) = 2 * A308542(n).

Extensions

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

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

Views

Author

Gus Wiseman, Jul 09 2019

Keywords

Examples

			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}

Links

Crossrefs

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.

Programs

  • Mathematica
    fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],Intersection[#,Times@@@Subsets[#,{2,n}]]=={}&]]],{n,0,10}]
  • PARI
    \\ See link for program file.
for(n=0, 30, print1(A325710(n), ", ")) \\ Andrew Howroyd, Aug 29 2019

Extensions

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

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

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
Showing 1-7 of 7 results.

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