On Weak Sidon Sequences

A sequence \(\{a_i : 1 \leq i \leq k\}\) of integers is a weak Sidon sequence if the sums \(a_i + a_j\) are all different for any \(i < j\). Let \(g(n)\) denote the maximum integer \(k\) for which there exists a weak Sidon sequence \(\{a_i : 1 \leq i \leq k\}\) such that \(1 \leq a_1 < \cdots < a_k \leq n\). Let the weak Sidon number \(G(k) = \text{min}\{n \mid g(n) = k\}\). In this note, \(g(n)\) and \(G(k)\) are studied, and \(g(n)\) is computed for \(n \leq 172\), based on which the weak Sidon number \(G(k)\) is determined for up to \(k = 17\).