A377419 -id:A377419 - cp's OEIS frontend

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

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

A382396 Number of minimum sized maximal subsets of {1..n} such that every pair of distinct elements has a different difference.

Original entry on oeis.org

1, 1, 1, 3, 1, 6, 14, 18, 14, 10, 4, 110, 172, 216, 226, 214, 184, 152, 116, 82, 50, 26, 10, 3696, 3904, 3942, 3768, 3504, 3016, 2548, 2060, 1598, 1170, 832, 538, 330, 196, 106, 52, 20, 10, 4, 2, 69610, 62594, 55294, 47610, 40502, 33538, 27254, 21544, 16764, 12676, 9258, 6534, 4516, 3042, 1990, 1254, 754, 448

Offset: 0

Andrew Howroyd, Mar 23 2025

nonn

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

			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(11) = 4 sets: {1,2,4,8}, {1,2,4,9}, {1,2,4,10}, {1,2,4,11}.
The a(42) = 2 sets: {10,18,19,25,30}, {13,18,24,25,33}.
See also the examples in A325879.

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

Cf. A143823, A325879, A377419, A382395, A382397 (minimum size of set).

a(n)={
  local(best,count); best=n+1;
  my(ismaxl(b,w)=for(k=1, n, if(!bittest(b,k) && !bitand(w,bitor(b,1< n, if(ismaxl(b,w),if(r

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<

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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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A382396 Number of minimum sized maximal subsets of {1..n} such that every pair of distinct elements has a different difference.

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

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PARI