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

A326016 Number of knapsack partitions of n such that no addition of one part up to the maximum is knapsack.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 3, 0, 0, 0, 1, 0, 8, 0, 8, 4, 3, 0, 11, 5, 3, 2, 5, 0, 29, 2, 9, 8, 20, 2
Offset: 1

Views

Author

Gus Wiseman, Jun 03 2019

Keywords

Comments

An integer partition is knapsack if every distinct submultiset has a different sum.
The Heinz numbers of these partitions are given by A326018.

Examples

			The initial terms count the following partitions:
  15: (5,4,3,3)
  21: (7,6,5,3)
  21: (7,5,3,3,3)
  24: (8,7,6,3)
  25: (7,5,5,4,4)
  27: (9,8,7,3)
  27: (9,7,6,5)
  27: (8,7,3,3,3,3)
  31: (10,8,6,6,1)
  33: (11,9,7,3,3)
  33: (11,8,5,5,4)
  33: (11,7,6,6,3)
  33: (11,7,3,3,3,3,3)
  33: (11,5,5,4,4,4)
  33: (10,9,8,3,3)
  33: (10,8,6,6,3)
  33: (10,8,3,3,3,3,3)

Crossrefs

Cf. A002033, A108917, A275972, A276024.
Cf. A325863, A325864, A325877, A325878, A325880, A326015, A326017, A326018.

Programs

  • Mathematica
    sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
ksQ[y_]:=Length[sums[Sort[y]]]==Times@@(Length/@Split[Sort[y]]+1)-1;
maxks[n_]:=Select[IntegerPartitions[n],ksQ[#]&&Select[Table[Sort[Append[#,i]],{i,Range[Max@@#]}],ksQ]=={}&];
Table[Length[maxks[n]],{n,30}]

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}]

A326015 Number of strict knapsack partitions of n such that no superset with the same maximum is knapsack.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 1, 1, 3, 2, 4, 4, 5, 3, 3, 4, 6, 2, 7, 6, 13, 9, 19, 16, 27, 21, 40, 33, 47, 37, 54, 48, 66, 51, 65, 65, 77, 64, 80, 71, 96, 60, 106, 95, 112, 93, 152, 114, 191, 131, 242, 192, 303, 210, 366, 300, 482, 352, 581, 450, 713, 539, 882, 689, 995
Offset: 1

Views

Author

Gus Wiseman, Jun 03 2019

Keywords

Comments

An integer partition is knapsack if every distinct submultiset has a different sum.
These are the subsets counted by A325867, ordered by sum rather than maximum.

Examples

			The a(1) = 1 through a(17) = 6 strict knapsack partitions (empty columns not shown):
  {1}  {2,1}  {3,1}  {3,2}  {4,2,1}  {5,2,1}  {4,3,2}  {6,3,1}  {5,4,2}
                                              {5,3,1}  {7,2,1}  {6,3,2}
                                              {6,2,1}           {6,4,1}
                                                                {7,3,1}
.
  {5,4,3}  {6,4,3}  {6,5,3}  {6,5,4}    {7,5,4}    {7,6,4}
  {7,3,2}  {6,5,2}  {8,5,1}  {7,6,2}    {9,4,3}    {9,5,3}
  {7,4,1}  {7,4,2}  {9,3,2}  {8,4,2,1}  {9,6,1}    {9,6,2}
  {8,3,1}  {7,5,1}                      {9,4,2,1}  {8,4,3,2}
           {9,3,1}                                 {9,5,2,1}
                                                   {10,4,2,1}

Links

Crossrefs

Cf. A002033, A108917, A275972, A276024.
Cf. A325863, A325864, A325877, A325878, A325880, A326016, A326017, A326018.

Programs

  • Mathematica
    ksQ[y_]:=UnsameQ@@Total/@Union[Subsets[y]]
maxsks[n_]:=Select[Select[IntegerPartitions[n],UnsameQ@@#&&ksQ[#]&],Select[Table[Append[#,i],{i,Complement[Range[Max@@#],#]}],ksQ]=={}&];
Table[Length[maxsks[n]],{n,30}]

A325863 Number of integer partitions of n such that every distinct non-singleton submultiset has a different sum.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 11, 15, 17, 24, 29, 31, 41, 51, 58, 67, 84, 91, 117, 117
Offset: 0

Views

Author

Gus Wiseman, May 31 2019

Keywords

Comments

A knapsack partition (A108917, A299702) is an integer partition such that every submultiset has a different sum. The one non-knapsack partition counted under a(4) is (2,1,1).

Examples

			The partition (2,1,1,1) has non-singleton submultisets {1,2} and {1,1,1} with the same sum, so (2,1,1,1) is not counted under a(5).
The a(1) = 1 through a(8) = 15 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (211)   (221)    (51)      (61)       (62)
                    (1111)  (311)    (222)     (322)      (71)
                            (11111)  (321)     (331)      (332)
                                     (411)     (421)      (422)
                                     (3111)    (511)      (431)
                                     (111111)  (2221)     (521)
                                               (4111)     (611)
                                               (1111111)  (2222)
                                                          (3311)
                                                          (5111)
                                                          (41111)
                                                          (11111111)
The 10 non-knapsack partitions counted under a(12):
  (7,6,1)
  (7,5,2)
  (7,4,3)
  (7,5,1,1)
  (7,4,2,1)
  (7,3,3,1)
  (7,3,2,2)
  (7,4,1,1,1)
  (7,2,2,2,1)
  (7,1,1,1,1,1,1,1)

Crossrefs

Dominates A108917.
Cf. A002033, A055212, A143823, A196723, A276024, A299702, A325856, A325862, A325864, A325865, A325866, A325867, A325877.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@Plus@@@Union[Subsets[#,{2,Length[#]}]]&]],{n,0,15}]

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

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

A326033 Number of knapsack partitions of n such that no addition of one part equal to an existing part is knapsack.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 3, 0, 0, 1, 1, 0, 8, 0, 8, 4, 3, 0, 11, 5, 3, 4, 5, 0, 30, 2, 9, 9, 20, 3, 37, 6, 18, 16, 37, 20, 71, 12, 37, 40
Offset: 1

Views

Author

Gus Wiseman, Jun 03 2019

Keywords

Comments

An integer partition is knapsack if every distinct submultiset has a different sum.

Examples

			The partition (10,8,6,6) is counted under a(30) because (10,10,8,6,6), (10,8,8,6,6), and (10,8,6,6,6) are not knapsack.

Crossrefs

Cf. A002033, A108917, A275972, A276024, A299702.
Cf. A325857, A325862, A325863, A325864, A325865, A326015, A326016, A326018.

Programs

  • Mathematica
    sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
ksQ[y_]:=Length[sums[Sort[y]]]==Times@@(Length/@Split[Sort[y]]+1)-1;
maxks[n_]:=Select[IntegerPartitions[n],ksQ[#]&&Select[Table[Sort[Append[#,i]],{i,Union[#]}],ksQ]=={}&];
Table[Length[maxks[n]],{n,30}]
Previous Showing 11-20 of 20 results.

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