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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A325857 Number of integer partitions of n such that every orderless pair of distinct parts has a different sum.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, 74, 97, 125, 165, 209, 269, 335, 428, 527, 664, 804, 1005, 1210, 1496, 1780, 2186, 2586, 3148, 3698, 4473, 5226, 6279, 7290, 8706, 10067, 11950, 13744, 16242, 18605, 21864, 24942, 29184, 33188, 38651, 43782, 50791, 57402, 66300, 74683, 86026, 96658
Offset: 0

Views

Author

Gus Wiseman, May 31 2019

Keywords

Examples

			The A000041(14) - a(14) = 10 partitions of 14 not satisfying the condition are:
  (6,5,2,1)
  (6,4,3,1)
  (5,4,3,2)
  (5,4,2,2,1)
  (4,4,3,2,1)
  (5,4,2,1,1,1)
  (4,3,3,2,1,1)
  (4,3,2,2,2,1)
  (4,3,2,2,1,1,1)
  (4,3,2,1,1,1,1,1)

Links

Crossrefs

The subset case is A196723.
The maximal case is A325878.
The integer partition case is A325857.
The strict integer partition case is A325877.
Heinz numbers of the counterexamples are given by A325991.
Cf. A002033, A103300, A108917, A143823, A196724, A325853, A325855, A325858, A325859, A325862.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@Plus@@@Subsets[Union[#],{2}]&]],{n,0,30}]

Extensions

Terms a(31) onward from Max Alekseyev, Sep 23 2023

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

A325880 Number of maximal subsets of {1..n} containing n such that every ordered pair of distinct elements has a different difference.

Original entry on oeis.org

1, 1, 2, 2, 4, 8, 8, 10, 18, 34, 50, 70, 78, 89, 120, 181, 277, 401, 561, 728, 867, 1031, 1219, 1537, 2013, 2684, 3581, 4973, 6435, 8124, 9974, 12054, 14057, 16890, 19783, 24102, 29539, 37247, 46301, 59825, 74556, 94064, 115057, 141068, 167521, 200790, 232798, 273734
Offset: 1

Views

Author

Gus Wiseman, Jun 02 2019

Keywords

Comments

Also the number of maximal subsets of {1..n} containing n such that every orderless pair of (not necessarily distinct) elements has a different sum.

Examples

			The a(2) = 1 through a(9) = 18 subsets:
  {1,2}  {1,3}  {1,2,4}  {1,2,5}  {1,2,6}  {2,3,7}    {3,5,8}    {4,6,9}
         {2,3}  {1,3,4}  {1,4,5}  {1,3,6}  {2,4,7}    {4,5,8}    {5,6,9}
                         {2,3,5}  {1,4,6}  {2,6,7}    {1,2,4,8}  {1,2,4,9}
                         {2,4,5}  {1,5,6}  {3,4,7}    {1,2,6,8}  {1,2,6,9}
                                  {2,3,6}  {4,5,7}    {1,3,4,8}  {1,2,7,9}
                                  {2,5,6}  {4,6,7}    {1,3,7,8}  {1,3,4,9}
                                  {3,4,6}  {1,2,5,7}  {1,5,6,8}  {1,3,8,9}
                                  {3,5,6}  {1,3,6,7}  {1,5,7,8}  {1,4,8,9}
                                                      {2,3,6,8}  {1,6,7,9}
                                                      {2,4,7,8}  {1,6,8,9}
                                                                 {2,3,5,9}
                                                                 {2,3,7,9}
                                                                 {2,4,5,9}
                                                                 {2,4,8,9}
                                                                 {2,6,7,9}
                                                                 {2,6,8,9}
                                                                 {3,4,7,9}
                                                                 {3,5,8,9}

Links

Crossrefs

Cf. A002033, A108917, A143823, A143824, A196723.
Cf. A325858, A325859, A325861, A325867, A325869, A325879, A325992.

Programs

  • Mathematica
    fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],MemberQ[#,n]&&UnsameQ@@Subtract@@@Subsets[Union[#],{2}]&]]],{n,0,10}]
  • PARI
    a(n)={
  my(ismaxl(b,w)=for(k=1, n, if(!bittest(b,k) && !bitand(w,bitor(b,1<= n, ismaxl(b,w),
         my(s=self()(k+1, b,w));
         b+=1<Andrew Howroyd, Mar 23 2025

Extensions

a(25) onwards from Andrew Howroyd, Mar 23 2025

A364537 Heinz numbers of integer partitions where some part is the difference of two consecutive parts.

Original entry on oeis.org

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 70, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 130, 132, 133, 138, 140, 144, 147, 150, 154, 156, 162, 165, 168, 174, 180, 186, 189, 192, 195, 198, 204, 210, 216, 222, 228, 231, 234, 240, 246, 252, 258
Offset: 1

Views

Author

Gus Wiseman, Aug 02 2023

Keywords

Comments

In other words, partitions whose parts are not disjoint from their first differences.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Examples

			The partition {3,4,5,7} with Heinz number 6545 has first differences (1,1,2) so is not in the sequence.
The terms together with their prime indices begin:
   6: {1,2}
  12: {1,1,2}
  18: {1,2,2}
  21: {2,4}
  24: {1,1,1,2}
  30: {1,2,3}
  36: {1,1,2,2}
  42: {1,2,4}
  48: {1,1,1,1,2}
  54: {1,2,2,2}
  60: {1,1,2,3}
  63: {2,2,4}
  65: {3,6}
  66: {1,2,5}
  70: {1,3,4}
  72: {1,1,1,2,2}
  78: {1,2,6}
  84: {1,1,2,4}
  90: {1,2,2,3}
  96: {1,1,1,1,1,2}

Crossrefs

For all differences of pairs the complement is A364347, counted by A364345.
For all differences of pairs we have A364348, counted by A363225.
Subsets of {1..n} of this type are counted by A364466, complement A364463.
These partitions are counted by A364467, complement A363260.
The strict case is A364536, complement A364464.
A050291 counts double-free subsets, complement A088808.
A323092 counts double-free partitions, ranks A320340.
A325325 counts partitions with distinct first differences.
Cf. A002865, A025065, A093971, A108917, A196723, A229816, A236912, A237113, A237667, A320347, A326083.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#],Differences[prix[#]]]!={}&]

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}]

A364673 Number of (necessarily strict) integer partitions of n containing all of their own first differences.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 1, 2, 2, 2, 5, 2, 2, 4, 2, 3, 6, 4, 4, 8, 4, 4, 10, 8, 7, 8, 13, 9, 15, 12, 13, 17, 20, 15, 31, 24, 27, 32, 33, 32, 50, 42, 45, 53, 61, 61, 85, 76, 86, 101, 108, 118, 137, 141, 147, 179, 184, 196, 222, 244, 257, 295, 324, 348, 380, 433
Offset: 0

Views

Author

Gus Wiseman, Aug 03 2023

Keywords

Examples

			The partition y = (12,6,3,2,1) has differences (6,3,1,1), and {1,3,6} is a subset of {1,2,3,6,12}, so y is counted under a(24).
The a(n) partitions for n = 1, 3, 6, 12, 15, 18, 21:
  (1)  (3)    (6)      (12)       (15)         (18)         (21)
       (2,1)  (4,2)    (8,4)      (10,5)       (12,6)       (14,7)
              (3,2,1)  (6,4,2)    (8,4,2,1)    (9,6,3)      (12,6,3)
                       (5,4,2,1)  (5,4,3,2,1)  (6,5,4,2,1)  (8,6,4,2,1)
                       (6,3,2,1)               (7,5,3,2,1)  (9,5,4,2,1)
                                               (8,4,3,2,1)  (9,6,3,2,1)
                                                            (10,5,3,2,1)
                                                            (6,5,4,3,2,1)

Links

Crossrefs

Containing all differences: A007862.
Containing no differences: A364464, strict complement A364536.
Containing at least one difference: A364467, complement A363260.
For subsets of {1..n} we have A364671, complement A364672.
A non-strict version is A364674.
For submultisets instead of subsets we have A364675.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A236912 counts sum-free partitions w/o re-used parts, complement A237113.
A325325 counts partitions with distinct first differences.
Cf. A002865, A025065, A196723, A229816, A237667, A320347, A363225, A364272, A364345, A364463, A364537, A370386.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&SubsetQ[#,-Differences[#]]&]],{n,0,30}]
  • Python
    from collections import Counter
def A364673_list(maxn):
    count = Counter()
    for i in range(maxn//3):
        A,f,i = [[(i+1, )]],0,0
        while f == 0:
            A.append([])
            for j in A[i]:
                for k in j:
                    x = j + (j[-1] + k, )
                    y = sum(x)
                    if y <= maxn:
                        A[i+1].append(x)
                        count.update({y})
            if len(A[i+1]) < 1: f += 1
            i += 1
    return [count[z]+1 for z in range(maxn+1)] # John Tyler Rascoe, Mar 09 2024

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}]

A325991 Heinz numbers of integer partitions such that not every orderless pair of distinct parts has a different sum.

Original entry on oeis.org

210, 420, 462, 630, 840, 858, 910, 924, 1050, 1155, 1260, 1326, 1386, 1470, 1680, 1716, 1820, 1848, 1870, 1890, 1938, 2100, 2145, 2310, 2470, 2520, 2574, 2622, 2652, 2730, 2772, 2926, 2940, 3150, 3234, 3315, 3360, 3432, 3465, 3570, 3640, 3696, 3740, 3780, 3876
Offset: 1

Views

Author

Gus Wiseman, Jun 02 2019

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The sequence of terms together with their prime indices begins:
   210: {1,2,3,4}
   420: {1,1,2,3,4}
   462: {1,2,4,5}
   630: {1,2,2,3,4}
   840: {1,1,1,2,3,4}
   858: {1,2,5,6}
   910: {1,3,4,6}
   924: {1,1,2,4,5}
  1050: {1,2,3,3,4}
  1155: {2,3,4,5}
  1260: {1,1,2,2,3,4}
  1326: {1,2,6,7}
  1386: {1,2,2,4,5}
  1470: {1,2,3,4,4}
  1680: {1,1,1,1,2,3,4}
  1716: {1,1,2,5,6}
  1820: {1,1,3,4,6}
  1848: {1,1,1,2,4,5}
  1870: {1,3,5,7}
  1890: {1,2,2,2,3,4}

Crossrefs

The subset case is A196723.
The maximal case is A325878.
The integer partition case is A325857.
The strict integer partition case is A325877.
Heinz numbers of the counterexamples are given by A325991.
Cf. A002033, A056239, A103300, A108917, A112798, A143823, A196724, A325853, A325855, A325858, A325859, A325862, A325992, A325993, A325994.

Programs

  • Mathematica
    Select[Range[1000],!UnsameQ@@Plus@@@Subsets[PrimePi/@First/@FactorInteger[#],{2}]&]
Previous Showing 11-20 of 33 results. Next

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