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, 150, 171

Offset

2,2

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

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

Table of n, a(n) for n=2..17.

A. Llado, Largest cliques in connected supermagic graphs, European Journal of Combinatorics 28 (2007), 2240--2247.

Example

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.

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 *)

Prog

(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

Crossrefs

See A003022, A004133, and A004135 for other versions.

Sequence in context: A239955 A346114 A332744 * A084672 A011910 A227590

Adjacent sequences: A345728 A345729 A345730 * A345732 A345733 A345734

Keyword

nonn,hard,more,nice

Author

Bernd Mulansky, Jun 25 2021

Status

approved