A382398 -id:A382398 - cp's OEIS frontend

A325878 Number of maximal subsets of {1..n} such that every orderless pair of distinct elements has a different sum.

1, 1, 1, 1, 4, 5, 8, 22, 40, 56, 78, 124, 222, 390, 616, 892, 1220, 1620, 2182, 3042, 4392, 6364, 9054, 12608, 16980, 22244, 28482, 36208, 45864, 58692, 75804, 98440, 128694, 168250, 218558, 281210, 357594, 449402, 560034, 693332, 853546, 1050118, 1293458, 1596144, 1975394

Offset: 0

Gus Wiseman, Jun 02 2019

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 0..60

The subset case is A196723.
The integer partition case is A325857.
The strict integer partition case is A325877.
Heinz numbers of the counterexamples are given by A325991.
Cf. A002033, A108917, A143823, A143824, A276024.
Cf. A325858, A325859, A325864, A325865, A325867, A325879, A325880, A382398.

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

a(n)={
   my(ismaxl(b,w)=for(k=1, n, if(!bittest(b,k) && !bitand(w,b< n, ismaxl(b,w),
         my(s=self()(k+1, r, b, w));
         if(!bitand(w,b<Andrew Howroyd, Mar 23 2025

a(21) onwards from Andrew Howroyd, Mar 23 2025

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<

cp's OEIS Frontend

A325878 Number of maximal subsets of {1..n} such that every orderless pair of distinct elements has a different sum.

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