A345731 Additive bases: a(n) is the least integer such that there is an n-element set of integers between 0 and a(n), the sums of pairs (of distinct elements) of which are distinct.

1, 2, 4, 7, 12, 18, 24, 34, 45, 57, 71, 86, 105, 126, 148

Offset: 2

Author

Bernd Mulansky, Jun 25 2021

Keywords

• nonn
• hard
• more
• nice

Comments

Such sets are known as weak Sidon sets, weak B_2 sets, or well-spread sequences.

n - 1 <= a(n) <= A003022(n). - Michael S. Branicky, Jun 25 2021

Examples

			a(6)=12 because 0-1-2-4-7-12 (0-5-8-10-11-12) resp. 0-1-2-6-9-12 (0-3-6-10-11-12) are shortest weak Sidon sets of size 6.
a(16)=148: [0, 3, 5, 6, 32, 49, 59, 68, 93, 106, 118, 126, 130, 134, 141, 148]. - _Zhao Hui Du_, Jul 27 2025
		

References

• Alison M. Marr and W. D. Wallis, Magic Graphs, Birkhäuser, 2nd ed., 2013. See Section 2.3.
• Xiaodong Xu, Meilian Liang, and Zehui Shao, On weak Sidon sequences, The Journal of Combinatorial Mathematics and Combinatorial Computing (2014), 107--113

Links

• A. Lladó, Largest cliques in connected supermagic graphs, European Journal of Combinatorics, Vol. 28, No. 8 (2007), 2240-2247.

Crossrefs

See A003022, A004133, and A004135 for other versions.

Programs

Mathematica

a[n_Integer?NonNegative] := Module[{k = n - 1}, While[SelectFirst[Subsets[Range[0, k - 1], {n - 1}], Length@Union[Plus @@@ Subsets[#~Join~{k}, {2}]] >= (n*(n - 1))/2 &] === Missing["NotFound"], k++]; k];
Table[a[n], {n, 2, 8}] (* Robert P. P. McKone, Nov 05 2023 *)

Python

from itertools import combinations, count
def a(n):
    for k in count(n-1):
        for c in combinations(range(k), n-1):
            c = c + (k,)
            ss = set()
            for s in combinations(c, 2):
                if sum(s) in ss: break
                else: ss.add(sum(s))
            if len(ss) == n*(n-1)//2: return k # use (k, c) for sets
print([a(n) for n in range(2, 9)]) # Michael S. Branicky, Jun 25 2021

Extensions

a(16) corrected and a(17) deleted by Zhao Hui Du, Jul 27 2025