A143824 -id:A143824 - cp's OEIS frontend

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

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

Gus Wiseman, Jun 02 2019

nonn

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.

			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}

Andrew Howroyd, Table of n, a(n) for n = 1..60

Cf. A002033, A108917, A143823, A143824, A196723.

Cf. A325858, A325859, A325861, A325867, A325869, A325879, A325992.

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}]

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

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

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

Gus Wiseman, Jun 01 2019

nonn

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

			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}

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..150 (terms 1..121 from Bert Dobbelaere)

Cf. A002033, A108917, A143823, A143824, A196723, A275972.

Cf. A325860, A325861, A325864, A325865, A325866, A325867, A325880.

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&)/@y];
Table[Length[fasmax[Select[Subsets[Range[n]],MemberQ[#,n]&&UnsameQ@@Plus@@@Subsets[#]&]]],{n,15}]

def f(p0, n, m, cm):
    full, t, p = True, 0, p0
    while p>k)&1)==0 and ((m<Bert Dobbelaere, Mar 07 2021

More terms from Bert Dobbelaere, Mar 07 2021

A325678 Maximum length of a composition of n such that every restriction to a subinterval has a different sum.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

Gus Wiseman, May 13 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

Also the maximum number of nonzero marks on a Golomb ruler of length n.

Run-lengths are A270813.
Cf. A000079, A007318, A103295, A108917, A143823, A169942.
Cf. A325466, A325676, A325677, A325679, A325681, A325683, A325687.

Table[Max[Length/@Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@ReplaceList[#,{_,s__,_}:>Plus[s]]&]],{n,0,15}]

a(n) + 1 = A143824(n + 1).

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

Gus Wiseman, Jun 01 2019

nonn

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

			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}

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..150

Cf. A108917, A143823, A143824, A196723, A275972.

Cf. A325860, A325864, A325865, A325867, A325877, A325878, A325880.

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

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

A308251 Number of subsets of {1,...,n + 1} containing n + 1 and such that all positive differences of distinct elements are distinct.

Original entry on oeis.org

1, 2, 3, 6, 9, 14, 21, 34, 49, 76, 101, 146, 205, 294, 397, 560, 747, 1028, 1341, 1810, 2343, 3178, 4051, 5370, 6921, 9014, 11361, 14838, 18719, 24082, 29953, 38220, 47663, 60550, 74619, 93848, 115961, 145320, 177549, 221676, 270335, 335124

Offset: 0

Gus Wiseman, May 17 2019

nonn

Also the number of subsets of {1...n} containing no positive differences of the elements and such that all such differences are distinct.

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

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..99 (terms 0..41 from Gus Wiseman, terms 42..80 from Vaclav Kotesovec)

Cf. A000079, A108917, A143823, A143824, A169942, A325676, A325677, A325683, A325687.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&UnsameQ@@Abs[Subtract@@@Subsets[#,{2}]]&]],{n,15}]

First differences of A143823. Partial sums of A169942.

A382395 Number of maximum sized subsets of {1..n} such that every pair of distinct elements has a different difference.

Original entry on oeis.org

1, 1, 1, 3, 2, 6, 14, 2, 10, 26, 60, 110, 4, 22, 68, 156, 320, 584, 8, 24, 80, 206, 504, 1004, 1910, 3380, 10, 34, 98, 282, 760, 1618, 3334, 6360, 11482, 2, 22, 70, 214, 540, 1250, 2718, 5712, 10910, 20418, 2, 12, 30, 90, 230, 562, 1228, 2690, 5550, 11260, 21164, 2, 4, 6, 10, 18

Offset: 0

Andrew Howroyd, Mar 23 2025

nonn

Also the number of maximum sized subsets of {1..n} such that every pair of (not necessarily distinct) elements has a different sum. In other words, a(n) is the number of Sidon sets with A143824(n) elements which are <= n.

			The a(0) = 1 set is {}.
The a(1) = 1 set is {1}.
The a(2) = 1 set is {1,2}.
The a(3) = 3 sets: {1,2}, {1,3}, {2,3}.
The a(4) = 2 sets: {1,2,4}, {1,3,4}.
The a(5) = 6 sets: {1,2,4}, {1,2,5}, {1,3,4}, {1,4,5}, {2,3,5}, {2,4,5}.
The a(6) = 14 sets: {1,2,4}, {1,2,5}, {1,2,6}, {1,3,4}, {1,3,6}, {1,4,5}, {1,4,6}, {1,5,6}, {2,3,5}, {2,3,6}, {2,4,5}, {2,5,6}, {3,4,6}, {3,5,6}.
The a(7) = 2 sets: {1,2,5,7}, {1,3,6,7}.

Wikipedia, Sidon sequence.
Index entries for sequences related to Golomb rulers.

Cf. A143823, A143824 (maximum size of set), A325879, A377410, A382396, A382398.

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

A326082 Number of maximal sets of pairwise indivisible divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 5, 2, 3, 3, 5, 2, 5, 2, 5, 3, 3, 2, 8, 3, 3, 4, 5, 2, 7, 2, 6, 3, 3, 3, 9, 2, 3, 3, 8, 2, 7, 2, 5, 5, 3, 2, 12, 3, 5, 3, 5, 2, 8, 3, 8, 3, 3, 2, 15, 2, 3, 5, 7, 3, 7, 2, 5, 3, 7, 2, 15, 2, 3, 5, 5, 3, 7, 2, 12, 5, 3, 2, 15, 3

Offset: 1

Gus Wiseman, Jun 05 2019

nonn

Depends only on prime signature.

The non-maximal case is A096827.

			The maximal sets of pairwise indivisible divisors of n = 1, 2, 4, 8, 12, 24, 30, 32, 36, 48, 60 are:
   1   1   1   1   1     1      1         1    1       1       1
       2   2   2   12    24     30        2    36      48      60
           4   4   2,3   2,3    5,6       4    2,3     2,3     2,15
               8   3,4   3,4    2,15      8    2,9     3,4     3,20
                   4,6   3,8    3,10      16   3,4     3,8     4,30
                         4,6    2,3,5     32   4,18    4,6     5,12
                         6,8    6,10,15        9,12    6,8     2,3,5
                         8,12                  12,18   3,16    3,4,5
                                               4,6,9   6,16    4,5,6
                                                       8,12    3,4,10
                                                       12,16   6,15,20
                                                       16,24   10,12,15
                                                               12,15,20
                                                               12,20,30
                                                               4,6,10,15

Cf. A001055, A051026, A067992, A096827, A143824, A285572, A285573, A303362, A305148, A305149, A316476, A325861, A326023, A326077.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Rest[Subsets[Divisors[n]]],stableQ[#,Divisible]&]]],{n,100}]

A377410 Maximum sum of a subset of {1..n} such that every pair of distinct elements has a different difference.

Original entry on oeis.org

0, 1, 3, 5, 8, 11, 14, 17, 21, 25, 29, 33, 37, 42, 47, 52, 57, 62, 67, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 151, 158, 165, 172, 179, 186, 193, 200, 207, 215, 223, 231, 239, 247, 255, 263, 271, 279, 287, 295, 303, 311, 319, 327, 335, 343, 351, 360, 369

Offset: 0

Andrew Howroyd, Oct 27 2024

nonn

Also the maximum sum of a subset of {1..n} such that every unordered pair of (not necessarily distinct) elements has a different sum. In other words, a(n) is the maximum sum of a Sidon set whose elements are <= n.

			a(0) = 0 = sum of {}.
a(1) = 1 = sum of {1}.
a(2) = 3 = sum of {1,2}.
a(3) = 5 = sum of {2,3}.
a(4) = 8 = sum of {1,3,4}.
a(5) = 11 = sum of {2,4,5}.
a(12) = 37 = sum of {6,8,11,12} or {5,9,11,12}.
a(20) = 78 = sum of {2,8,12,17,19,20}.
See also the examples in A143823.

Wikipedia, Sidon sequence.
Index entries for sequences related to Golomb rulers.

Cf. A143823, A143824 (maximum size of set), A377419.

a(n)={
   my(recurse(k,b,w)=
      if(k > n, 0,
         my(s=self()(k+1, b, w));
         b+=1<

def a(n):
    def recurse(k, b, w):
        if k > n: return 0
        s = recurse(k+1, b, w)
        b += (1<Michael S. Branicky, Oct 27 2024 after Andrew Howroyd

Name edited by Andrew Howroyd, Mar 24 2025

A382397 Minimum size of a maximal subset of {1..n} such that every pair of distinct elements has a different difference.

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

Andrew Howroyd, Mar 23 2025

Also the minimum size of a maximal subset of {1..n} such that every pair of (not necessarily distinct) elements has a different sum.
a(n) is the minimum size of a set enumerated by A325879(n).
Number of occurrences of k: 1, 1, 3,  6, 12, 20, ...
Maximum n having a(n) = k:  0, 1, 4, 10, 22, 42, ...
There are insufficient known terms in either of the above to distinguish from other sequences.

Thomas Bloom, Does there exist a maximal Sidon set A ⊂ {1, ..., N} of size O(N^(1/3))?, Erdős Problems.
Terence Tao, Erdős problem database, see no. 156.
Wikipedia, Sidon sequence.
Index entries for sequences related to Golomb rulers.

Cf. A143824 (maximum size of set), A325879, A377419 (minimum sum), A382396.

a(n)={
  my(ismaxl(b,w)=for(k=1, n, if(!bittest(b,k) && !bitand(w,bitor(b,1< n, if(ismaxl(b,w),0,n+1),
         my(s=self()(k+1, b,w));
         b+=1<

cp's OEIS Frontend

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

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

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A325678 Maximum length of a composition of n such that every restriction to a subinterval has a different sum.

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

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A308251 Number of subsets of {1,...,n + 1} containing n + 1 and such that all positive differences of distinct elements are distinct.

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A382395 Number of maximum sized subsets of {1..n} such that every pair of distinct elements has a different difference.

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A326082 Number of maximal sets of pairwise indivisible divisors of n.

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A377410 Maximum sum of a subset of {1..n} such that every pair of distinct elements has a different difference.

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