On Weak Sidon Sequences

Abstract

A sequence $\{a_i:1\le i\le k\}$ 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 $\{a_i:1\le i\le k\}$ such that $1\le a_1<\cdots <a_k\le n$. 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 $n\le 172$, based on which the weak Sidon number $G(k)$ is determined for up to $k=17$.