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 11-20 of 20 results.

A325865 Number of maximal subsets of {1..n} of which every subset has a different sum.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 14, 23, 27, 40, 64, 104, 180, 275, 399, 554, 679, 872, 1117, 1431, 1920, 2520, 3530, 4751, 6644, 8855, 12021, 15461, 19939, 25109, 31656, 38750, 46204, 55650, 65942, 78045, 91304, 106592, 124761, 145701, 172343, 201217, 238739, 280601, 339746, 400394
Offset: 0

Views

Author

Gus Wiseman, Jun 01 2019

Keywords

Examples

			The a(1) = 1 through a(6) = 14 subsets:
  {1}  {1,2}  {1,2}  {1,3}    {1,2,4}  {1,2,4}
              {1,3}  {1,2,4}  {1,2,5}  {1,2,5}
              {2,3}  {2,3,4}  {1,3,5}  {1,2,6}
                              {2,3,4}  {1,3,5}
                              {2,4,5}  {1,3,6}
                              {3,4,5}  {1,4,6}
                                       {2,3,4}
                                       {2,3,6}
                                       {2,4,5}
                                       {2,5,6}
                                       {3,4,5}
                                       {3,4,6}
                                       {3,5,6}
                                       {4,5,6}

Links

Crossrefs

Cf. A002033, A108917, A143823, A196723, A275972.
Cf. A325860, A325864, A325866, A325867, A325877, A325878, A325879, A325880.

Programs

  • Mathematica
    fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&)/@y];
Table[Length[fasmax[Select[Subsets[Range[n]],UnsameQ@@Plus@@@Subsets[#]&]]],{n,0,10}]
  • PARI
    a(n)={
  my(ismaxl(w)=for(k=1, n, if(!bitand(w,w< n, ismaxl(w),
         my(s=self()(k+1, b,w));
         if(!bitand(w,w<Andrew Howroyd, Mar 23 2025

Extensions

a(18) onwards from Andrew Howroyd, Mar 23 2025

A325868 Number of subsets of {1..n} containing n such that every ordered pair of distinct elements has a different quotient.

Original entry on oeis.org

1, 2, 4, 6, 14, 24, 52, 84, 120, 240, 548, 688, 1784, 2600, 4236, 5796, 16200, 17568, 49968, 55648, 101360, 176792, 433736, 430032, 728784, 1360928, 2304840, 2990856, 8682912, 7877376, 25243200, 27946656, 46758912, 81457248, 121546416, 114388320, 442583952
Offset: 1

Views

Author

Gus Wiseman, Jun 02 2019

Keywords

Examples

			The a(1) = 1 through a(5) = 14 subsets:
  {1}  {2}    {3}      {4}      {5}
       {1,2}  {1,3}    {1,4}    {1,5}
              {2,3}    {2,4}    {2,5}
              {1,2,3}  {3,4}    {3,5}
                       {1,3,4}  {4,5}
                       {2,3,4}  {1,2,5}
                                {1,3,5}
                                {1,4,5}
                                {2,3,5}
                                {2,4,5}
                                {3,4,5}
                                {1,2,3,5}
                                {1,3,4,5}
                                {2,3,4,5}

Links

Crossrefs

Cf. A025582, A108917, A143823, A196723, A196724.
Cf. A325853, A325854, A325858, A325860, A325861, A325864, A325869.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&UnsameQ@@Divide@@@Subsets[#,{2}]&]],{n,10}]

Extensions

a(21)-a(37) from Fausto A. C. Cariboni, Oct 16 2020

A325867 Number of maximal subsets of {1..n} containing n such that every subset has a different sum.

Original entry on oeis.org

1, 1, 2, 2, 4, 8, 10, 12, 17, 34, 45, 77, 99, 136, 166, 200, 238, 328, 402, 660, 674, 1166, 1331, 1966, 2335, 3286, 3527, 4762, 5383, 6900, 7543, 9087, 10149, 12239, 13569, 16452, 17867, 22869, 23977, 33881, 33820, 43423, 48090, 68683, 67347, 95176, 97917, 131666, 136205
Offset: 1

Views

Author

Gus Wiseman, Jun 01 2019

Keywords

Comments

These are maximal strict knapsack partitions (A275972, A326015) organized by maximum rather than sum.

Examples

			The a(1) = 1 through a(8) = 12 subsets:
  {1}  {1,2}  {1,3}  {1,2,4}  {1,2,5}  {1,2,6}  {1,2,7}    {1,3,8}
              {2,3}  {2,3,4}  {1,3,5}  {1,3,6}  {1,3,7}    {1,5,8}
                              {2,4,5}  {1,4,6}  {1,4,7}    {5,7,8}
                              {3,4,5}  {2,3,6}  {1,5,7}    {1,2,4,8}
                                       {2,5,6}  {2,3,7}    {1,4,6,8}
                                       {3,4,6}  {2,4,7}    {2,3,4,8}
                                       {3,5,6}  {2,6,7}    {2,4,5,8}
                                       {4,5,6}  {4,5,7}    {2,4,7,8}
                                                {4,6,7}    {3,4,6,8}
                                                {3,5,6,7}  {3,6,7,8}
                                                           {4,5,6,8}
                                                           {4,6,7,8}

Links

Crossrefs

Cf. A002033, A108917, A143823, A143824, A196723, A275972.
Cf. A325860, A325861, A325864, A325865, A325866, A325867, A325880.

Programs

  • Mathematica
    fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&)/@y];
Table[Length[fasmax[Select[Subsets[Range[n]],MemberQ[#,n]&&UnsameQ@@Plus@@@Subsets[#]&]]],{n,15}]
  • Python
    def f(p0, n, m, cm):
    full, t, p = True, 0, p0
    while p>k)&1)==0 and ((m<Bert Dobbelaere, Mar 07 2021

Extensions

More terms from Bert Dobbelaere, Mar 07 2021

A325869 Number of maximal subsets of {1..n} containing n such that every pair of distinct elements has a different quotient.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 6, 6, 20, 32, 29, 57, 83, 113, 183, 373, 233, 549, 360
Offset: 1

Views

Author

Gus Wiseman, Jun 02 2019

Keywords

Examples

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

Crossrefs

Cf. A025582, A067992, A108917, A143823, A196723, A196724.
Cf. A325853, A325854, A325858, A325860, A325861, A325864, A325868.

Programs

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

A326078 Number of subsets of {2..n} containing all of their integer quotients > 1.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 24, 48, 72, 144, 216, 432, 552, 1104, 1656, 2592, 3936, 7872, 10056, 20112, 26688, 42320, 63480, 126960, 154800, 309600, 464400, 737568, 992160, 1984320, 2450880, 4901760, 6292800, 10197312, 15295968, 26241696, 32947488, 65894976, 98842464, 161587872, 205842528
Offset: 0

Views

Author

Gus Wiseman, Jun 05 2019

Keywords

Comments

These sets are closed under taking the quotient of two distinct divisible terms.

Examples

			The a(6) = 24 subsets:
  {}  {2}  {2,3}  {2,3,4}  {2,3,4,5}  {2,3,4,5,6}
      {3}  {2,4}  {2,3,5}  {2,3,4,6}
      {4}  {2,5}  {2,3,6}  {2,3,5,6}
      {5}  {3,4}  {2,4,5}
      {6}  {3,5}  {3,4,5}
           {4,5}  {4,5,6}
           {4,6}
           {5,6}

Crossrefs

Cf. A007865, A051026, A054519, A067992, A103580, A325860, A325994, A326023, A326076, A326079, A326081.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[2,n]],SubsetQ[#,Divide@@@Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]]&]],{n,0,10}]
  • PARI
    a(n)={
    my(lim=vector(n, k, sqrtint(k)));
    my(accept(b, k)=for(i=2, lim[k], if(k%i ==0 && bittest(b,i) != bittest(b,k/i), return(0))); 1);
    my(recurse(k, b)=
      my(m=1);
      for(j=max(2*k,n\2+1), min(2*k+1,n), if(accept(b,j), m*=2));
      k++;
      m*if(k > n\2, 1, (self()(k, b) + if(accept(b, k), self()(k, b + (1<Andrew Howroyd, Aug 30 2019

Formula

For n > 0, a(n) = A326023(n) - 1.
For n > 0, a(n) = A326079(n)/2.

Extensions

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

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.

A325866 Number of subsets of {1..n} containing n such that every subset has a different sum.

Original entry on oeis.org

1, 2, 3, 6, 9, 14, 20, 35, 44, 76, 96, 139, 179, 257, 312, 483, 561, 793, 970, 1459, 1535, 2307, 2619, 3503, 4130, 5478, 5973, 8165, 9081, 11666, 13176, 17738, 18440, 24778, 26873, 35187, 38070, 49978, 51776, 72457, 74207, 92512, 102210, 135571, 136786, 179604
Offset: 1

Views

Author

Gus Wiseman, Jun 01 2019

Keywords

Comments

These are strict knapsack partitions (A275972) organized by maximum rather than sum.

Examples

			The a(1) = 1 through a(6) = 14 subsets:
  {1}  {2}    {3}    {4}      {5}      {6}
       {1,2}  {1,3}  {1,4}    {1,5}    {1,6}
              {2,3}  {2,4}    {2,5}    {2,6}
                     {3,4}    {3,5}    {3,6}
                     {1,2,4}  {4,5}    {4,6}
                     {2,3,4}  {1,2,5}  {5,6}
                              {1,3,5}  {1,2,6}
                              {2,4,5}  {1,3,6}
                              {3,4,5}  {1,4,6}
                                       {2,3,6}
                                       {2,5,6}
                                       {3,4,6}
                                       {3,5,6}
                                       {4,5,6}

Links

Crossrefs

Cf. A108917, A143823, A143824, A196723, A275972.
Cf. A325860, A325864, A325865, A325867, A325877, A325878, A325880.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&UnsameQ@@Plus@@@Subsets[#]&]],{n,10}]

Extensions

a(18)-a(46) from Alois P. Heinz, Jun 03 2019

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

A338006 Maximal size of a subset of {1..n} such that every pair of (not necessarily distinct) elements has a different product.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 17, 17, 18, 18, 19, 19, 19, 19, 20, 20, 20, 21, 22, 23, 24, 24, 24, 24, 25, 25, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 31, 31, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 36, 36, 37, 37, 37, 37, 38
Offset: 1

Views

Author

Fausto A. C. Cariboni, Oct 06 2020

Keywords

Links

Crossrefs

Cf. A325860, A347498.

Formula

a(n) = max {k >= 1; A347498(k) <= n}. - Pontus von Brömssen, Sep 09 2021

Extensions

a(52)-a(83) from Bert Dobbelaere, Oct 18 2020
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