A003022 -id:A003022 - cp's OEIS frontend

A227590 a(n) = A003022(n)+1 with a(1)=1.

1, 2, 4, 7, 12, 18, 26, 35, 45, 56, 73, 86, 107, 128, 152, 178, 200, 217, 247, 284, 334, 357, 373, 426, 481, 493, 554, 586

Offset: 1

Jens Voß, Jul 17 2013

Since A003022 is the most important sequence dealing with Golomb rulers, it seems best to define this sequence in terms of that one.
Original name was: Maximum label within a minimal labeling of 2 identical n-sided dice yielding the most possible sums. For example, two hexahedra labeled (1, 3, 8, 14, 17, 18) yield the 21 possible sums 2, 4, 6, 9, 11, 15, 16, 17, 18, 19, 20, 21, 22, 25, 26, 28, 31, 32, 34, 35, 36. No more sums can be obtained by different labelings, and no labeling with labels < 18 yields 21 possible sums. Therefore a(6) = 18.
Bounded above by A005282. - James Wilcox, Jul 27 2013
Minimum greatest integer in a set of n positive integers with all the differences between any two of its elements being different. - Javier Múgica, Jul 31 2015

Thomas Bloom, Problem 30, Problem 43, Problem 155, and Problem 861, Erdős Problems.
Isaac Mammel, William Smith, and Carl Yerger, Ramsey Theory on the Integer Grid: The "L" Problem, arXiv:2502.05162 [math.CO], 2025. See p. 12.
Terence Tao, Erdős problem database, see nos. 30, 43, 155, 861.

Cf. A003022.

Column k=2 of array A227588.

More terms from James Wilcox, Jul 27 2013
Entry revised by N. J. A. Sloane, Apr 08 2016
a(28) from A003022 added by Michel Marcus, Feb 10 2025

A270813 First differences of A003022.

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 10, 11, 17, 13, 21, 21, 24, 26, 22, 17, 30, 37, 50, 23, 16, 53, 55, 12, 61

Offset: 2

N. J. A. Sloane, Apr 08 2016

Surprisingly irregular.

Cf. A003022.

A078106 a(n) = A003022(n) - A000217(n-1).

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 6, 8, 10, 17, 19, 28, 36, 46, 57, 63, 63, 75, 93, 123, 125, 119, 149, 180, 167, 202, 207

Offset: 2

Michael Gilleland, Nov 24 2002

nonn

Previous name was: Number of distances not measured by the shortest Golomb ruler with n marks (i.e., length of shortest Golomb ruler with n marks minus number of distances measured by that ruler).

J. C. P Miller, Difference bases. Three problems in additive number theory. Computers in number theory (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969), pp. 299-322. Academic Press, London, 1971. MR0316269 (47 #4817) - From N. J. A. Sloane, Jun 05 2012
Rhys Price Jones, Gracelessness, Proc. 10th S.-E. Conf. Combin., Graph Theory and Computing, 1979, pp. 547-552. - From N. J. A. Sloane, Jun 05 2012

Index entries for sequences related to Golomb rulers

Cf. A000217, A003022, A106683.

a(n) = A003022(n) - A000217(n-1).

Made consistent with A003022 by Peter Munn, Jul 23 2019

a(24)-a(28) computed using A003022 by Floris P. van Doorn, Mar 06 2023

A005282 Mian-Chowla sequence (a B_2 sequence): a(1) = 1; for n>1, a(n) = smallest number > a(n-1) such that the pairwise sums of elements are all distinct.

Original entry on oeis.org

1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 290, 361, 401, 475, 565, 593, 662, 775, 822, 916, 970, 1016, 1159, 1312, 1395, 1523, 1572, 1821, 1896, 2029, 2254, 2379, 2510, 2780, 2925, 3155, 3354, 3591, 3797, 3998, 4297, 4433, 4779, 4851

Offset: 1

N. J. A. Sloane and Simon Plouffe

An alternative definition is to start with 1 and then continue with the least number such that all pairwise differences of distinct elements are all distinct. - Jens Voß, Feb 04 2003. [However, compare A003022 and A227590. - N. J. A. Sloane, Apr 08 2016]
Rachel Lewis points out [see link] that S, the sum of the reciprocals of this sequence, satisfies 2.158435 <= S <= 2.158677. Similarly, the sum of the squares of reciprocals of this sequence converges to approximately 1.33853369 and the sum of the cube of reciprocals of this sequence converges to approximately 1.14319352. - Jonathan Vos Post, Nov 21 2004; comment changed by N. J. A. Sloane, Jan 02 2020
Let S denote the reciprocal sum of a(n). Then 2.158452685 <= S <= 2.158532684. - Raffaele Salvia, Jul 19 2014
From Thomas Ordowski, Sep 19 2014: (Start)
Known estimate: n^2/2 + O(n) < a(n) < n^3/6 + O(n^2).
Conjecture: a(n) ~ n^3 / log(n)^2. (End)

			The second term is 2 because the 3 pairwise sums 1+1=2, 1+2=3, 2+2=4 are all distinct.
The third term cannot be 3 because 1+3 = 2+2. But it can be 4, since 1+4=5, 2+4=6, 4+4=8 are distinct and distinct from the earlier sums 1+1=2, 1+2=3, 2+2=4.

P. Erdős and R. Graham, Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique (1980).
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.20.2.
R. K. Guy, Unsolved Problems in Number Theory, E28.
A. M. Mian and S. D. Chowla, On the B_2-sequences of Sidon, Proc. Nat. Acad. Sci. India, A14 (1944), 3-4.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..5818 (terms less than 2*10^9)
Thomas Bloom, Problem 340, Erdős Problems.
Yin Choi Cheng, Greedy Sidon sets for linear forms, J. Num. Theor. (2024).
Rachel Lewis, Mian-Chowla and B2 sequences, 1999. [Thanks to _Steven Finch_ for providing this document. Included with permission. - _N. J. A. Sloane_, Jan 02 2020]
Kevin O'Bryant, B_h-Sets and Rigidity, arXiv:2312.10910 [math.NT], 2023.
Raffaele Salvia, Table of n, a(n) for n=1...25000
R. Salvia, A New Lower Bound for the Distinct Distance Constant, J. Int. Seq. 18 (2015) # 15.4.8.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)
Eric Weisstein's World of Mathematics, B2 Sequence.
Eric Weisstein's World of Mathematics, Chowla Sequence.
Zhang Zhen-Xiang, A B_2-sequence with larger reciprocal sum, Math. Comp. 60 (1993), 835-839.
Index entries for B_2 sequences.

Row 2 of A347570.
Cf. A051788, A080200 (for differences between terms).
Different from A046185. Cf. A011185.
See also A003022, A227590.
A259964 has a greater sum of reciprocals.
Cf. A002858.

import Data.Set (Set, empty, insert, member)
a005282 n = a005282_list !! (n-1)
a005282_list = sMianChowla [] 1 empty where
   sMianChowla :: [Integer] -> Integer -> Set Integer -> [Integer]
   sMianChowla sums z s | s' == empty = sMianChowla sums (z+1) s
                        | otherwise   = z : sMianChowla (z:sums) (z+1) s
      where s' = try (z:sums) s
            try :: [Integer] -> Set Integer -> Set Integer
            try []     s                      = s
            try (x:sums) s | (z+x) `member` s = empty
                           | otherwise        = try sums $ insert (z+x) s
-- Reinhard Zumkeller, Mar 02 2011

a[1]:= 1: P:= {2}: A:= {1}:
for n from 2 to 100 do
  for t from a[n-1]+1 do
    Pt:= map(`+`,A union {t},t);
    if Pt intersect P = {} then break fi
  od:
  a[n]:= t;
  A:= A union {t};
  P:= P union Pt;
od:
seq(a[n],n=1..100); # Robert Israel, Sep 21 2014

t = {1}; sms = {2}; k = 1; Do[k++; While[Intersection[sms, k + t] != {}, k++]; sms = Join[sms, t + k, {2 k}]; AppendTo[t, k], {49}]; t (* T. D. Noe, Mar 02 2011 *)

A005282_vec(N, A=[1], U=[0], D(A, n=#A)=vector(n-1, k, A[n]-A[n-k]))={ while(#A2 && U=U[k-1..-1]);A} \\ M. F. Hasler, Oct 09 2019

aupto(L)= my(S=vector(L), A=[1]); for(i=2, L, for(j=1, #A, if(S[i-A[j]], next(2))); for(j=1, #A, S[i-A[j]]=1); A=concat(A, i)); A \\ Ruud H.G. van Tol, Jun 30 2025

from itertools import count, islice
def A005282_gen(): # generator of terms
    aset1, aset2, alist = set(), set(), []
    for k in count(1):
        bset2 = {k<<1}
        if (k<<1) not in aset2:
            for d in aset1:
                if (m:=d+k) in aset2:
                    break
                bset2.add(m)
            else:
                yield k
                alist.append(k)
                aset1.add(k)
                aset2 |= bset2
A005282_list = list(islice(A005282_gen(),30)) # Chai Wah Wu, Sep 05 2023

a(n) = A025582(n) + 1.

a(n) = (A034757(n)+1)/2.

Examples added by N. J. A. Sloane, Jun 01 2008

A333224 Number of distinct positive consecutive subsequence-sums of the k-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 3, 3, 1, 3, 2, 4, 3, 4, 4, 4, 1, 3, 3, 5, 3, 5, 4, 5, 3, 4, 5, 5, 5, 5, 5, 5, 1, 3, 3, 5, 2, 5, 5, 6, 3, 6, 3, 6, 5, 6, 5, 6, 3, 4, 6, 6, 5, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 1, 3, 3, 5, 3, 6, 6, 7, 3, 5, 5, 7, 4, 6, 6, 7, 3, 6, 4, 7, 5, 7, 6

Offset: 0

Gus Wiseman, Mar 18 2020

nonn

The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.

			The composition (4,3,1,2) has positive subsequence-sums 1, 2, 3, 4, 6, 7, 8, 10, so a(550) = 8.

Dominated by A124770.
Compositions where every subinterval has a different sum are counted by A169942 and A325677 and ranked by A333222. The case of partitions is counted by A325768 and ranked by A325779.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack partitions are counted by A108917 and A325592 and ranked by A299702.
Strict knapsack partitions are counted by A275972 and ranked by A059519 and A301899.
Knapsack compositions are counted by A325676 and A325687 and ranked by A333223. The case of partitions is counted by A325769 and ranked by A325778, for which the number of distinct consecutive subsequences is given by A325770.
Allowing empty subsequences gives A333257.
Cf. A000120, A003022, A029931, A048793, A066099, A070939, A103295, A124767, A143823, A295235, A325680, A333217.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[ReplaceList[stc[n],{_,s__,_}:>Plus[s]]]],{n,0,100}]

a(n) = A333257(n) - 1.

A124771 Number of distinct subsequences for compositions in standard order.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 4, 4, 2, 4, 3, 6, 4, 6, 6, 5, 2, 4, 4, 6, 4, 6, 6, 8, 4, 6, 6, 9, 6, 9, 8, 6, 2, 4, 4, 6, 3, 7, 7, 8, 4, 7, 4, 9, 7, 8, 9, 10, 4, 6, 7, 9, 7, 9, 8, 12, 6, 9, 9, 12, 8, 12, 10, 7, 2, 4, 4, 6, 4, 7, 7, 8, 4, 6, 6, 10, 6, 10, 10, 10, 4, 7, 6, 10, 6, 8, 9, 12, 7, 10, 9, 12, 10, 12, 12, 12, 4

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.
From Vladimir Shevelev, Dec 18 2013: (Start)
Every number in binary is a concatenation of parts of the form 10...0 with k>=0 zeros. For example, 5=(10)(1), 11=(10)(1)(1), 7=(1)(1)(1). We call d>0 a c-divisor of m, if d consists of some consecutive parts of m taking from the left to the right. Note that, to d=0 corresponds an empty set of parts. So it is natural to consider 0 as a c-divisor of every m. For example, 5=(10)(1) is a divisor of 23=(10)(1)(1)(1). Analogously, 1,2,3,7,11,23 are c-divisors of 23. But 6=(1)(10) is not a c-divisor of 23.
One can prove a one-to-one correspondence between distinct subsequences for  composition no. n in standard order and c-divisors of n. So, the sequence lists also numbers of c-divisors of nonnegative integers.
(End)
These are contiguous subsequences, or restrictions to a subinterval. The case for all subsequences is A334299. - Gus Wiseman, Jun 02 2020

			Composition number 11 is 2,1,1; the subsequences are (empty); 1; 2; 1,1; 2,1; 2,1,1; so a(11) = 6.
The table starts:
1
2
1 2
1 3 3 3
Let n=11=(10)(1)(1). We have the following c-divisors of 11: 0,1,2,3,5,11. Thus a(11)=6. Note, that 3=(1)(1) is not a c-divisor of 13=(1)(10)(1) since, although it contains parts of 3=(1)(1), but in non-consecutive order. The c-divisors of 13 are 0,1,2,5,6,13. So, a(13)=6.
From _Gus Wiseman_, Jun 01 2020: (Start)
The c-divisors of n are given in column n below:
  0  0  0  0  0  0  0  0  0  0  0   0   0   0   0   0   0   0   0
     1  2  1  4  1  1  1  8  1  2   1   1   1   1   1   16  1   2
           3     2  2  3     4  10  2   4   2   2   3       8   4
                 5  6  7     9      3   12  5   3   7       17  18
                                    5       6   6   15
                                    11      13  14
(End)

Alois P. Heinz, Rows n = 0..14, flattened

Cf. A000005, A011782 (row lengths), A066099, A114994, A233249, A233312.
Not allowing empty subsequences gives A124770.
Dominates A333257.
The case for not just contiguous subsequences is A334299.
Positions of first appearances are A335279.
Compositions where every subinterval has a different sum are A333222.
Knapsack compositions are A333223.
Cf. A000120, A003022, A029931, A070939, A108917, A124767, A325680, A325770, A333224, A334967, A334968.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[ReplaceList[stc[n],{_,s___,_}:>{s}]]],{n,0,100}] (* Gus Wiseman, Jun 01 2020 *)

a(n) = A124770(n) + 1.
From Vladimir Shevelev, Dec 18 2013: (Start)
a(2^n) = 2. Note that in concatenation representations of integers in binary, numbers {2^k}, k>=0, play the role of primes. So the formula is an analog of A000005(prime(n))=2.
a(2^n-1) = n+1; for n>=2, a(2^n+1) = 4.
For c-equivalent numbers n_1 and n_2 (i.e., differed only by order of parts) we have a(n_1) = a(n_2). For example, a(24)=a(17)=4. If the canonical representation of n is n=(1)^k_1[*](10)^k_2[*](100)^k_3[*]... , where [*] denotes operation of concatenation (cf. A233569), then a(n)<=(k_1+1)*(k_2+1)*...
(End)

A333223 Numbers k such that every distinct consecutive subsequence of the k-th composition in standard order has a different sum.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 17, 18, 19, 20, 21, 24, 26, 28, 31, 32, 33, 34, 35, 36, 40, 41, 42, 48, 50, 56, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 80, 81, 84, 85, 88, 96, 98, 100, 104, 106, 112, 120, 127, 128, 129, 130, 131, 132, 133

Offset: 1

Gus Wiseman, Mar 17 2020

nonn

The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.

			The list of terms together with the corresponding compositions begins:
    0: ()            21: (2,2,1)           65: (6,1)
    1: (1)           24: (1,4)             66: (5,2)
    2: (2)           26: (1,2,2)           67: (5,1,1)
    3: (1,1)         28: (1,1,3)           68: (4,3)
    4: (3)           31: (1,1,1,1,1)       69: (4,2,1)
    5: (2,1)         32: (6)               70: (4,1,2)
    6: (1,2)         33: (5,1)             71: (4,1,1,1)
    7: (1,1,1)       34: (4,2)             72: (3,4)
    8: (4)           35: (4,1,1)           73: (3,3,1)
    9: (3,1)         36: (3,3)             74: (3,2,2)
   10: (2,2)         40: (2,4)             80: (2,5)
   12: (1,3)         41: (2,3,1)           81: (2,4,1)
   15: (1,1,1,1)     42: (2,2,2)           84: (2,2,3)
   16: (5)           48: (1,5)             85: (2,2,2,1)
   17: (4,1)         50: (1,3,2)           88: (2,1,4)
   18: (3,2)         56: (1,1,4)           96: (1,6)
   19: (3,1,1)       63: (1,1,1,1,1,1)     98: (1,4,2)
   20: (2,3)         64: (7)              100: (1,3,3)

Distinct subsequences are counted by A124770 and A124771.
A superset of A333222, counted by A169942, with partition case A325768.
These compositions are counted by A325676.
A version for partitions is A325769, with Heinz numbers A325778.
The number of distinct positive subsequence-sums is A333224.
The number of distinct subsequence-sums is A333257.
Numbers whose binary indices are a strict knapsack partition are A059519.
Knapsack partitions are counted by A108917, with strict case A275972.
Golomb subsets are counted by A143823.
Heinz numbers of knapsack partitions are A299702.
Maximal Golomb rulers are counted by A325683.
Cf. A000120, A003022, A029931, A048793, A066099, A070939, A103295 A325779, A233564, A325680, A325687, A325770, A333217.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@Total/@Union[ReplaceList[stc[#],{_,s__,_}:>{s}]]&]

A333222 Numbers k such that every restriction of the k-th composition in standard order to a subinterval has a different sum.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 12, 16, 17, 18, 20, 24, 32, 33, 34, 40, 41, 48, 50, 64, 65, 66, 68, 69, 70, 72, 80, 81, 88, 96, 98, 104, 128, 129, 130, 132, 133, 134, 144, 145, 160, 161, 176, 192, 194, 196, 208, 256, 257, 258, 260, 261, 262, 264, 265, 268, 272, 274

Offset: 1

Gus Wiseman, Mar 17 2020

nonn

Also numbers whose binary indices together with 0 define a Golomb ruler.

The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.

			The list of terms together with the corresponding compositions begins:
    0: ()        41: (2,3,1)    130: (6,2)      262: (6,1,2)
    1: (1)       48: (1,5)      132: (5,3)      264: (5,4)
    2: (2)       50: (1,3,2)    133: (5,2,1)    265: (5,3,1)
    4: (3)       64: (7)        134: (5,1,2)    268: (5,1,3)
    5: (2,1)     65: (6,1)      144: (3,5)      272: (4,5)
    6: (1,2)     66: (5,2)      145: (3,4,1)    274: (4,3,2)
    8: (4)       68: (4,3)      160: (2,6)      276: (4,2,3)
    9: (3,1)     69: (4,2,1)    161: (2,5,1)    288: (3,6)
   12: (1,3)     70: (4,1,2)    176: (2,1,5)    289: (3,5,1)
   16: (5)       72: (3,4)      192: (1,7)      290: (3,4,2)
   17: (4,1)     80: (2,5)      194: (1,5,2)    296: (3,2,4)
   18: (3,2)     81: (2,4,1)    196: (1,4,3)    304: (3,1,5)
   20: (2,3)     88: (2,1,4)    208: (1,2,5)    320: (2,7)
   24: (1,4)     96: (1,6)      256: (9)        321: (2,6,1)
   32: (6)       98: (1,4,2)    257: (8,1)      324: (2,4,3)
   33: (5,1)    104: (1,2,4)    258: (7,2)      328: (2,3,4)
   34: (4,2)    128: (8)        260: (6,3)      352: (2,1,6)
   40: (2,4)    129: (7,1)      261: (6,2,1)    384: (1,8)

Wikipedia, Golomb ruler

A subset of A233564.
Also a subset of A333223.
These compositions are counted by A169942 and A325677.
The number of distinct nonzero subsequence-sums is A333224.
The number of distinct subsequence-sums is A333257.
Lengths of optimal Golomb rulers are A003022.
Inequivalent optimal Golomb rulers are counted by A036501.
Complete rulers are A103295, with perfect case A103300.
Knapsack partitions are counted by A108917, with strict case A275972.
Distinct subsequences are counted by A124770 and A124771.
Golomb subsets are counted by A143823.
Heinz numbers of knapsack partitions are A299702.
Knapsack compositions are counted by A325676.
Maximal Golomb rulers are counted by A325683.
Cf. A000120, A029931, A048793, A066099, A070939, A228351, A295235, A325678, A325680, A325768, A325779, A333217.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,300],UnsameQ@@ReplaceList[stc[#],{_,s__,_}:>Plus[s]]&]

A143824 Size of the largest subset {x(1),x(2),...,x(k)} of {1,2,...,n} with the property that all differences |x(i)-x(j)| are distinct.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12

Offset: 0

John W. Layman, Sep 02 2008

When the set {x(1),x(2),...,x(k)} satisfies the property that all differences |x(i)-x(j)| are distinct (or alternately, all the sums are distinct), then it is called a Sidon set. So a(n) is the maximum cardinality of a dense Sidon subset of {1,2,...,n}. - Sayan Dutta, Aug 29 2024
See A143823 for the number of subsets of {1, 2, ..., n} with the required property.
See A003022 (and A227590) for the values of n such that a(n+1) > a(n). - Boris Bukh, Jul 28 2013
Can be formulated as an integer linear program: maximize sum {i = 1 to n} z[i] subject to z[i] + z[j] - 1 <= y[i,j] for all i < j, sum {i = 1 to n - d} y[i,i+d] <= 1 for d = 1 to n - 1, z[i] in {0,1} for all i, y[i,j] in {0,1} for all i < j. - Rob Pratt, Feb 09 2010
If the zeroth term is removed, the run-lengths are A270813 with 1 prepended. - Gus Wiseman, Jun 07 2019

			For n = 4, {1, 2, 4} is a subset of {1, 2, 3, 4} with distinct differences 2 - 1 = 1, 4 - 1 = 3, 4 - 2 = 2 between pairs of elements and no larger set has the required property; so a(4) = 3.
From _Gus Wiseman_, Jun 07 2019: (Start)
Examples of subsets realizing each largest size are:
   0: {}
   1: {1}
   2: {1,2}
   3: {2,3}
   4: {1,3,4}
   5: {2,4,5}
   6: {3,5,6}
   7: {1,3,6,7}
   8: {2,4,7,8}
   9: {3,5,8,9}
  10: {4,6,9,10}
  11: {5,7,10,11}
  12: {1,4,5,10,12}
  13: {2,5,6,11,13}
  14: {3,6,7,12,14}
  15: {4,7,8,13,15}
(End)

N. J. A. Sloane, Table of n, a(n) for n = 0..500
Thomas Bloom, Problem 30, Problem 43, Problem 155, and Problem 861, Erdős Problems.
Kevin O'Bryant, A complete annotated bibliography of work related to Sidon sequences, Electron. J. Combin., DS11, Dynamic Surveys (2004), 39 pp.
Terence Tao, Erdős problem database, see nos. 30, 43, 155, 861.
Wikipedia, Sidon sequence.
Wikipedia, Golomb ruler
Balogh, J., Füredi, Z., & Roy, S. (2023), An Upper Bound on the Size of Sidon Sets, The American Mathematical Monthly, 130(5), 437-445.

Cf. A143823, A003002, A003022, A227590.

Cf. A108917, A169942, A270813, A325676, A325677, A325678, A325683, A325687.

Table[Length[Last[Select[Subsets[Range[n]],UnsameQ@@Subtract@@@Subsets[#,{2}]&]]],{n,0,15}] (* Gus Wiseman, Jun 07 2019 *)

For n > 1, a(n) = A325678(n - 1) + 1. - Gus Wiseman, Jun 07 2019
From Sayan Dutta, Aug 29 2024: (Start)
a(n) < n^(1/2) + 0.998*n^(1/4) for sufficiently large n (see Balogh et. al. link).
It is conjectured by Erdos (for $500) that a(n) < n^(1/2) + o(n^e) for all e>0. (End)

More terms from Rob Pratt, Feb 09 2010
a(41)-a(60) from Alois P. Heinz, Sep 14 2011
More terms and b-file from N. J. A. Sloane, Apr 08 2016 using data from A003022.

A124770 Number of distinct nonempty subsequences for compositions in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 3, 3, 1, 3, 2, 5, 3, 5, 5, 4, 1, 3, 3, 5, 3, 5, 5, 7, 3, 5, 5, 8, 5, 8, 7, 5, 1, 3, 3, 5, 2, 6, 6, 7, 3, 6, 3, 8, 6, 7, 8, 9, 3, 5, 6, 8, 6, 8, 7, 11, 5, 8, 8, 11, 7, 11, 9, 6, 1, 3, 3, 5, 3, 6, 6, 7, 3, 5, 5, 9, 5, 9, 9, 9, 3, 6, 5, 9, 5, 7, 8, 11, 6, 9, 8, 11, 9, 11, 11, 11, 3, 5, 6, 8, 5, 9

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. - Gus Wiseman, Apr 03 2020

			Composition number 11 is 2,1,1; the nonempty subsequences are 1; 2; 1,1; 2,1; 2,1,1; so a(11) = 5.
The table starts:
  0
  1
  1 2
  1 3 3 3
  1 3 2 5 3 5 5 4
  1 3 3 5 3 5 5 7 3 5 5 8 5 8 7 5
From _Gus Wiseman_, Apr 03 2020: (Start)
If the k-th composition in standard order is c, then we say that the STC-number of c is k. The STC-numbers of the distinct subsequences of the composition with STC-number k are given in column k below:
  1  2  1  4  1  1  1  8  1  2   1   1   1   1   1   16  1   2   1   2
        3     2  2  3     4  10  2   4   2   2   3       8   4   4   4
              5  6  7     9      3   12  6   3   7       17  18  3   20
                                 5       5   6   15              9
                                 11      13  14                  19
(End)

Alois P. Heinz, Rows n = 0..14, flattened

Row lengths are A011782.
Allowing empty subsequences gives A124771.
Dominates A333224, the version counting subsequence-sums instead of subsequences.
Compositions where every restriction to a subinterval has a different sum are counted by A169942 and A325677 and ranked by A333222. The case of partitions is counted by A325768 and ranked by A325779.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack compositions are counted by A325676 and A325687 and ranked by A333223. The case of partitions is counted by A325769 and ranked by A325778, for which the number of distinct consecutive subsequences is given by A325770.
Cf. A000120, A003022, A029931, A048793, A066099, A070939, A103295, A108917, A143823, A325680.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[ReplaceList[stc[n],{_,s__,_}:>{s}]]],{n,0,100}] (* Gus Wiseman, Apr 03 2020 *)

a(n) = A124771(n) - 1. - Gus Wiseman, Apr 03 2020

cp's OEIS Frontend

A227590 a(n) = A003022(n)+1 with a(1)=1.

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A270813 First differences of A003022.

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A078106 a(n) = A003022(n) - A000217(n-1).

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A005282 Mian-Chowla sequence (a B_2 sequence): a(1) = 1; for n>1, a(n) = smallest number > a(n-1) such that the pairwise sums of elements are all distinct.

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A333224 Number of distinct positive consecutive subsequence-sums of the k-th composition in standard order.

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A124771 Number of distinct subsequences for compositions in standard order.

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A333223 Numbers k such that every distinct consecutive subsequence of the k-th composition in standard order has a different sum.

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A333222 Numbers k such that every restriction of the k-th composition in standard order to a subinterval has a different sum.

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