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.

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A364671 Number of subsets of {1..n} containing all of their own first differences.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 23, 34, 58, 96, 171, 302, 565, 1041, 1969, 3719, 7105, 13544, 25999, 49852, 95949, 184658, 356129, 687068, 1327540, 2566295, 4966449, 9617306, 18640098, 36150918, 70166056, 136272548, 264844111, 515036040, 1002211421, 1951345157, 3801569113
Offset: 0

Views

Author

Gus Wiseman, Aug 04 2023

Keywords

Examples

			The subset {1,2,4,5,10,14} has differences (1,2,1,5,4) so is counted under a(14).
The a(0) = 1 through a(5) = 14 subsets:
  {}  {}   {}     {}       {}         {}
      {1}  {1}    {1}      {1}        {1}
           {2}    {2}      {2}        {2}
           {1,2}  {3}      {3}        {3}
                  {1,2}    {4}        {4}
                  {1,2,3}  {1,2}      {5}
                           {2,4}      {1,2}
                           {1,2,3}    {2,4}
                           {1,2,4}    {1,2,3}
                           {1,2,3,4}  {1,2,4}
                                      {1,2,3,4}
                                      {1,2,3,5}
                                      {1,2,4,5}
                                      {1,2,3,4,5}

Links

Crossrefs

For differences of all strict pairs we have A054519, for partitions A007862.
For "disjoint" instead of "subset" we have A364463, partitions A363260.
For "non-disjoint" we have A364466, partitions A364467 (strict A364536).
The complement is counted by A364672, partitions A364673, A364674, A364675.
First differences of terms are A364752, complement A364753.
Cf. A151897, A196723, A237667, A237668, A325325, A326083, A363225, A364345, A364464, A364537.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]], SubsetQ[#,Differences[#]]&]], {n,0,10}]

Extensions

More terms from Rémy Sigrist, Aug 06 2023

A196719 Number of subsets of {1..n} (including empty set) such that the pairwise GCDs of elements are all distinct.

Original entry on oeis.org

1, 2, 4, 7, 11, 16, 24, 31, 40, 52, 68, 79, 102, 115, 140, 175, 201, 218, 265, 284, 336, 396, 446, 469, 547, 599, 662, 742, 837, 866, 1034, 1065, 1153, 1275, 1370, 1511, 1719, 1756, 1869, 2030, 2244, 2285, 2613, 2656, 2865, 3236, 3394, 3441, 3780, 3921, 4232
Offset: 0

Views

Author

Alois P. Heinz, Oct 05 2011

Keywords

Examples

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

Links

Crossrefs

Cf. A143823, A196720, A196721, A196722, A196723, A196724.

Programs

  • Maple
    b:= proc(n, s) local sn, m;
      m:= nops(s);
      sn:= [s[], n];
      `if`(n<1, 1, b(n-1, s) +`if`(m*(m+1)/2 = nops(({seq(seq(
       igcd(sn[i], sn[j]), j=i+1..m+1), i=1..m)})), b(n-1, sn), 0))
    end:
a:= proc(n) option remember;
      b(n-1, [n]) +`if`(n=0, 0, a(n-1))
    end:
seq(a(n), n=0..50);
  • Mathematica
    b[n_, s_] := b[n, s] = With[{m = Length[s], sn = Append[s, n]}, If[n<1, 1, b[n-1, s] + If[m*(m+1)/2 == Length[ Union @ Flatten @ Table[ Table[ GCD[ sn[[i]], sn[[j]]], {j, i+1, m+1}], {i, 1, m}]], b[n-1, sn], 0]]];
a[n_] := a[n] = b[n-1, {n}] + If[n == 0, 0, a[n-1]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 06 2017, translated from Maple *)

A196720 Number of subsets of {1..n} (including empty set) such that the pairwise GCDs of elements are not distinct.

Original entry on oeis.org

1, 2, 4, 8, 13, 25, 33, 61, 81, 116, 140, 256, 282, 530, 606, 692, 823, 1551, 1653, 3173, 3391, 3805, 4177, 8049, 8345, 11524, 12508, 15294, 16204, 31692, 32048, 63280, 70834, 77224, 82048, 91686, 93597, 185245, 196109, 212359, 218223, 432495, 436031, 867647
Offset: 0

Views

Author

Alois P. Heinz, Oct 05 2011

Keywords

Comments

All pairwise GCDs of each subset are equal if there are any.
a(n) >= A084422(n).

Examples

			a(5) = 25: {}, {1}, {2}, {3}, {4}, {5}, {1,2}, {1,3}, {1,4}, {1,5}, {2,3}, {2,4}, {2,5}, {3,4}, {3,5}, {4,5}, {1,2,3}, {1,2,5}, {1,3,4}, {1,3,5}, {1,4,5}, {2,3,5}, {3,4,5}, {1,2,3,5}, {1,3,4,5}.

Links

Crossrefs

Cf. A143823, A196719, A196721, A196722, A196723, A196724.

Programs

  • Maple
    b:= proc(n, s) local sn, m;
      m:= nops(s);
      sn:= [s[], n];
      `if`(n<1, 1, b(n-1, s) +`if`(1 >= nops(({seq(seq(
           igcd(sn[i], sn[j]), j=i+1..m+1), i=1..m)})), b(n-1, sn), 0))
    end:
a:= proc(n) option remember;
      b(n-1, [n]) +`if`(n=0, 0, a(n-1))
    end:
seq(a(n), n=0..20);
  • Mathematica
    b[n_, s_] := b[n, s] = With[{m = Length[s], sn = Append[s, n]}, If[n<1, 1, b[n-1, s] + If[1 >= Length[ Union @ Flatten @ Table[ Table[ GCD[ sn[[i]], sn[[j]]], {j, i+1, m+1}], {i, 1, m}]], b[n-1, sn], 0]]];
a[n_] := a[n] = b[n-1, {n}] + If[n == 0, 0, a[n-1]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 06 2017, translated from Maple *)

A196721 Number of subsets of {1..n} (including empty set) such that the pairwise LCMs of elements are all distinct.

Original entry on oeis.org

1, 2, 4, 8, 14, 28, 42, 84, 132, 236, 352, 704, 920, 1840, 2736, 3816, 5700, 11400, 15384, 30768, 39552, 54656, 81672, 163344, 196176, 362656, 542304, 930352, 1195168, 2390336, 2914304, 5828608, 8513920, 11674848, 17490432, 23484224, 28058816, 56117632, 84100800
Offset: 0

Views

Author

Alois P. Heinz, Oct 05 2011

Keywords

Examples

			a(4) = 14: {}, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,3,4}, {2,3,4}.

Links

Crossrefs

Cf. A143823, A196719, A196720, A196722, A196723, A196724.

Programs

  • Maple
    b:= proc(n, s) local sn, m;
      m:= nops(s);
      sn:= [s[], n];
      `if`(n<1, 1, b(n-1, s) +`if`(m*(m+1)/2 = nops(({seq(seq(
       ilcm(sn[i], sn[j]), j=i+1..m+1), i=1..m)})), b(n-1, sn), 0))
    end:
a:= proc(n) option remember;
      b(n-1, [n]) +`if`(n=0, 0, a(n-1))
    end:
seq(a(n), n=0..10);
  • Mathematica
    b[n_, s_] := b[n, s] = Module[{sn, m}, m = Length[s]; sn = Append[s, n]; If[n < 1, 1, b[n - 1, s] + If[m*(m + 1)/2 == Length @ Union @ Flatten @ Table[LCM [sn[[i]], sn[[j]]], {i, 1, m}, {j, i+1, m+1}], b[n-1, sn], 0]]]; a[n_] := a[n] = b[n-1, {n}] + If[n == 0, 0, a[n-1]]; Table[ Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 02 2017, translated from Maple *)

Extensions

Terms a(31) and beyond from Fausto A. C. Cariboni, Oct 18 2020

A196722 Number of subsets of {1..n} (including empty set) such that the pairwise LCMs of elements are not distinct.

Original entry on oeis.org

1, 2, 4, 7, 11, 16, 23, 30, 38, 47, 58, 69, 83, 96, 111, 128, 144, 161, 181, 200, 223, 246, 269, 292, 319, 344, 371, 398, 429, 458, 496, 527, 559, 594, 629, 668, 708, 745, 784, 825, 870, 911, 962, 1005, 1052, 1102, 1149, 1196, 1248, 1297, 1349, 1402, 1457, 1510
Offset: 0

Views

Author

Alois P. Heinz, Oct 05 2011

Keywords

Comments

All pairwise LCMs of each subset are equal if there are any.

Examples

			A(6) = 23: {}, {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}, {4,5}, {4,6}, {5,6}, {2,3,6}.

Links

Crossrefs

Cf. A143823, A196719, A196720, A196721, A196723, A196724.

Programs

  • Maple
    b:= proc(n, s) local sn, m;
      m:= nops(s);
      sn:= [s[], n];
      `if`(n<1, 1, b(n-1, s) +`if`(1 >= nops(({seq(seq(
           ilcm(sn[i], sn[j]), j=i+1..m+1), i=1..m)})), b(n-1, sn), 0))
    end:
a:= proc(n) option remember;
      b(n-1, [n]) +`if`(n=0, 0, a(n-1))
    end:
seq(a(n), n=0..50);
  • Mathematica
    b[n_, s_] := b[n, s] = Module[{sn, m}, m = Length[s]; sn = Append[s, n]; If[n<1, 1, b[n-1, s] + If[1 >= Length @ Union @ Flatten @ Table[ LCM[ sn[[i]], sn[[j]]], {i, 1, m}, {j, i+1, m+1}], b[n-1, sn], 0]]];
a[n_] := a[n] = b[n-1, {n}] + If[n == 0, 0, a[n-1]];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Apr 12 2017, translated from Maple *)

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

A364752 Number of subsets of {1..n} containing n and all first differences.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 9, 11, 24, 38, 75, 131, 263, 476, 928, 1750, 3386, 6439, 12455, 23853, 46097, 88709, 171471, 330939, 640472, 1238755, 2400154, 4650857, 9022792, 17510820, 34015138, 66106492, 128571563, 250191929, 487175381, 949133736, 1850223956, 3608650389
Offset: 0

Views

Author

Gus Wiseman, Aug 06 2023

Keywords

Examples

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

Links

Crossrefs

Partial sums are A364671, complement A364672.
The complement is counted by A364753.
A054519 counts subsets containing differences, A326083 containing sums.
A364463 counts subsets disjoint from differences, complement A364466.
A364673 counts partitions containing differences, A364674, A364675.
Cf. A151897, A196723, A237668, A325325, A363225, A364345, A364464, A364537.

Programs

  • Mathematica
    Table[If[n==0,1,Length[Select[Subsets[Range[n]], MemberQ[#,n]&&SubsetQ[#,Differences[#]]&]]],{n,0,10}]

Extensions

More terms from Rémy Sigrist, Aug 06 2023

A364465 Number of subsets of {1..n} with all different first differences of elements.

Original entry on oeis.org

1, 2, 4, 7, 13, 22, 36, 61, 99, 156, 240, 381, 587, 894, 1334, 1967, 2951, 4370, 6406, 9293, 13357, 18976, 27346, 39013, 55437, 78154, 109632, 152415, 210801, 293502, 406664, 561693, 772463, 1058108, 1441796, 1956293, 2639215, 3579542, 4835842, 6523207
Offset: 0

Views

Author

Gus Wiseman, Jul 30 2023

Keywords

Examples

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

Links

Crossrefs

For all differences of pairs of elements we have A196723
For partitions instead of subsets we have A325325, strict A320347.
For subset-sums we have A325864, for partitions A108917, A275972.
A007318 counts subsets by length.
A053632 counts subsets by sum.
A363260 counts partitions disjoint from differences, complement A364467.
A364463 counts subsets disjoint from differences, complement A364466.
Cf. A000009, A008289, A011782, A236912, A320348, A325857, A325877, A325878, A326083, A364345, A364346.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],UnsameQ@@Differences[#]&]],{n,0,10}]

Extensions

More terms from Rémy Sigrist, Aug 06 2023

A364753 Number of subsets of {1..n} containing n but not containing all first differences.

Original entry on oeis.org

0, 0, 0, 2, 4, 12, 23, 53, 104, 218, 437, 893, 1785, 3620, 7264, 14634, 29382, 59097, 118617, 238291, 478191, 959867, 1925681, 3863365, 7748136, 15538461, 31154278, 62458007, 125194936, 250924636, 502855774, 1007635332, 2018912085, 4044775367, 8102759211, 16230735448, 32509514412, 65110826347
Offset: 0

Views

Author

Gus Wiseman, Aug 06 2023

Keywords

Comments

In other words, subsets containing both n and some element that is not the difference of two consecutive elements.

Examples

			The a(1) = 0 through a(6) = 23 subsets:
  .  .  {1,3}  {1,4}    {1,5}      {1,6}
        {2,3}  {3,4}    {2,5}      {2,6}
               {1,3,4}  {3,5}      {4,6}
               {2,3,4}  {4,5}      {5,6}
                        {1,2,5}    {1,2,6}
                        {1,3,5}    {1,3,6}
                        {1,4,5}    {1,4,6}
                        {2,3,5}    {1,5,6}
                        {2,4,5}    {2,3,6}
                        {3,4,5}    {2,5,6}
                        {1,3,4,5}  {3,4,6}
                        {2,3,4,5}  {3,5,6}
                                   {4,5,6}
                                   {1,2,5,6}
                                   {1,3,4,6}
                                   {1,3,5,6}
                                   {1,4,5,6}
                                   {2,3,4,6}
                                   {2,3,5,6}
                                   {2,4,5,6}
                                   {3,4,5,6}
                                   {1,3,4,5,6}
                                   {2,3,4,5,6}

Crossrefs

Partial sums are A364672, complement A364671.
The complement is counted by A364752.
A054519 counts subsets containing differences, A326083 containing sums.
A364463 counts subsets disjoint from differences, complement A364466.
A364673, A364674, A364675 count partitions containing differences.
Cf. A151897, A196723, A237668, A325325, A364345, A364464, A364537.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&!SubsetQ[#,Differences[#]]&]],{n,0,10}]

Extensions

More terms from Giorgos Kalogeropoulos, Aug 07 2023

A382398 Number of maximum sized subsets of {1..n} such that every pair of distinct elements has a different sum.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 8, 22, 2, 14, 40, 102, 214, 4, 24, 92, 236, 564, 1148, 4, 18, 90, 270, 694, 1558, 2, 6, 24, 76, 252, 632, 1554, 3282, 6820, 12942, 6, 24, 84, 246, 664, 1562, 3442, 7084, 14336, 27202, 50520, 2, 26, 88, 294, 704, 1716, 3708, 8028, 16108, 31466, 58320, 107136, 4, 20, 54
Offset: 0

Views

Author

Andrew Howroyd, Mar 23 2025

Keywords

Examples

			The a(1) = 1 through a(6) = 8 subsets:
  {1}  {1,2}  {1,2,3}  {1,2,3}  {1,2,3,5}  {1,2,3,5}
                       {1,2,4}  {1,3,4,5}  {1,2,3,6}
                       {1,3,4}             {1,2,4,6}
                       {2,3,4}             {1,3,4,5}
                                           {1,3,5,6}
                                           {1,4,5,6}
                                           {2,3,4,6}
                                           {2,4,5,6}
Compare the above examples with A325878.

Crossrefs

Cf. A039836 (maximum size), A196723, A325878, A382395.

Programs

  • PARI
    a(n)={
   local(best,count);
   my(recurse(k,r,b,w)=
      if(k > n, if(r>=best, if(r>n,best=r;count=0); count++),
         self()(k+1, r, b, w);
         if(!bitand(w,b<
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