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 31-39 of 39 results.

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

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

A326498 Number of maximal subsets of {1..n} containing no sums of distinct elements.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 11, 16, 20, 32, 53, 78, 107, 149, 206, 292, 391, 556, 782, 1062, 1451, 1929, 2564, 3404, 4431, 5853, 7672, 9999, 12973, 16922, 22194, 28655, 36734, 47036, 60375, 76866, 97892, 123627, 157008, 196633, 248221, 311442, 390859, 488327, 610685
Offset: 0

Views

Author

Gus Wiseman, Jul 09 2019

Keywords

Examples

			The a(1) = 1 through a(7) = 16 subsets:
  {1}  {1,2}  {1,2}  {1,3}    {1,2,4}  {1,2,4}    {1,2,4}
              {1,3}  {1,2,4}  {1,2,5}  {1,2,5}    {1,2,5}
              {2,3}  {2,3,4}  {1,3,5}  {1,2,6}    {1,2,6}
                              {2,3,4}  {1,3,5}    {1,2,7}
                              {2,4,5}  {1,3,6}    {1,3,6}
                              {3,4,5}  {1,4,6}    {1,4,6}
                                       {2,3,4}    {1,4,7}
                                       {2,3,6}    {2,3,4}
                                       {2,4,5}    {2,4,5}
                                       {2,5,6}    {2,4,7}
                                       {3,4,5,6}  {2,5,6}
                                                  {1,3,5,7}
                                                  {2,3,6,7}
                                                  {3,4,5,6}
                                                  {3,5,6,7}
                                                  {4,5,6,7}

Links

Crossrefs

Subsets without sums of distinct elements are A151897.
Maximal sum-free subsets are A121269.
Subsets with sums are A326083.
Maximal subsets without products of distinct elements are A325710.
Maximal subsets without sums or products of distinct elements are A326025.
Cf. A007865, A103580, A326117, A326495, A326497.

Programs

  • Mathematica
    fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],Intersection[#,Plus@@@Subsets[#,{2,n}]]=={}&]]],{n,0,10}]
  • PARI
    \\ See link for program file.
for(n=0, 25, print1(A326498(n), ", ")) \\ Andrew Howroyd, Aug 29 2019

Extensions

a(16)-a(40) from Andrew Howroyd, Aug 29 2019
a(41)-a(44) from Jinyuan Wang, Oct 11 2020

A365071 Number of subsets of {1..n} containing n such that no element is a sum of distinct other elements. A variation of non-binary sum-free subsets without re-usable elements.

Original entry on oeis.org

0, 1, 2, 3, 6, 9, 15, 23, 40, 55, 94, 132, 210, 298, 476, 644, 1038, 1406, 2149, 2965, 4584, 6077, 9426, 12648, 19067, 25739, 38958, 51514, 78459, 104265, 155436, 208329, 312791, 411886, 620780, 823785, 1224414, 1631815, 2437015, 3217077, 4822991
Offset: 0

Views

Author

Gus Wiseman, Aug 26 2023

Keywords

Comments

The complement is counted by A365069. The binary version is A364755, complement A364756. For re-usable parts we have A288728, complement A365070.

Examples

			The subset {1,3,4,6} has 4 = 1 + 3 so is not counted under a(6).
The subset {2,3,4,5,6} has 6 = 2 + 4 and 4 = 1 + 3 so is not counted under a(6).
The a(0) = 0 through a(6) = 15 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}
                                          {3,4,5,6}

Links

Crossrefs

First differences of A151897.
The version with re-usable parts is A288728 first differences of A007865.
The binary version is A364755, first differences of A085489.
The binary complement is A364756, first differences of A088809.
The complement is counted by A365069, first differences of A364534.
The complement w/ re-usable parts is A365070, first differences of A093971.
A108917 counts knapsack partitions, strict A275972.
A124506 counts combination-free subsets, differences of A326083.
A364350 counts combination-free strict partitions, complement A364839.
A365046 counts combination-full subsets, differences of A364914.
Partitions: A236912, A237113, A237668, A364532, A364272, A364349, A364913.
Cf. A050291, A095944, A103580, A324741, A326080, A326117, A341507, A364533.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]], MemberQ[#,n]&&Intersection[#, Total/@Subsets[#,{2,Length[#]}]]=={}&]], {n,0,10}]

Formula

a(n) + A365069(n) = 2^(n-1).
First differences of A151897.

Extensions

a(14) onwards added (using A151897) by Andrew Howroyd, Jan 13 2024

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

A326115 Number of maximal double-free subsets of {1..n}.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 4, 4, 6, 6, 12, 12, 12, 12, 24, 24, 32, 32, 64, 64, 64, 64, 128, 128, 192, 192, 384, 384, 384, 384, 768, 768, 960, 960, 1920, 1920, 1920, 1920, 3840, 3840, 5760, 5760, 11520, 11520, 11520, 11520, 23040, 23040, 30720, 30720
Offset: 0

Views

Author

Gus Wiseman, Jun 06 2019

Keywords

Comments

A set is double-free if no element is twice any other element.

Examples

			The a(1) = 1 through a(9) = 6 sets:
  {1}  {1}  {13}  {23}   {235}   {235}   {2357}   {13457}  {134579}
       {2}  {23}  {134}  {1345}  {256}   {2567}   {13578}  {135789}
                                 {1345}  {13457}  {14567}  {145679}
                                 {1456}  {14567}  {15678}  {156789}
                                                  {23578}  {235789}
                                                  {25678}  {256789}

Links

Crossrefs

Cf. A000931, A007865, A050291, A103580, A120641, A308546, A320340, A323092.
Cf. A325865, A326020, A326022, A326083.

Programs

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

Formula

From Charlie Neder, Jun 11 2019: (Start)
a(n) = Product {k < n/2} A000931(8+floor(log_2(n/(2k+1)))).
a(2k+1) = a(2k), a(8k+4) = a(8k+3). (End)

Extensions

a(16)-a(49) from Charlie Neder, Jun 11 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

A358392 Number of nonempty subsets of {1, 2, ..., n} with GCD equal to 1 and containing the sum of any two elements whenever it is at most n.

Original entry on oeis.org

1, 1, 2, 3, 7, 9, 19, 27, 46, 63, 113, 148, 253, 345, 539, 734, 1198, 1580, 2540, 3417, 5233, 7095, 11190, 14720, 22988, 31057, 47168, 63331, 98233, 129836, 200689, 269165, 406504, 546700, 838766, 1108583, 1700025, 2281517, 3437422, 4597833, 7023543, 9308824, 14198257, 18982014, 28556962
Offset: 1

Views

Author

Max Alekseyev, Nov 13 2022

Keywords

Comments

Also, the number of distinct numerical semigroups that are generated by some subset of {1, 2, ..., n} and have a finite complement in the positive integers.

Links

Crossrefs

Inverse Moebius transform of A103580.
Cf. A007865, A050291, A051026, A085489, A139384, A151897, A308546, A326020, A326076, A326080, A326083, A326114.

Formula

a(n) = Sum_{k=1..n} moebius(k) * A103580(floor(n/k)).

A364841 Number of subsets S of {1..n} containing no element equal to the sum of a k-multiset of elements of S, for any 2 <= k <= |S|.

Original entry on oeis.org

1, 2, 3, 6, 9, 15, 21, 34, 49, 75, 105
Offset: 0

Views

Author

Gus Wiseman, Aug 15 2023

Keywords

Examples

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

Crossrefs

Cf. A007865, A085489, A103580, A151897, A236912, A326020, A326080, A326083, A364349, A364534.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]], Intersection[#,Join@@Table[Total/@Tuples[#,k], {k,2,Length[#]}]]=={}&]],{n,0,10}]
Previous Showing 31-39 of 39 results.

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