OPEN
Let $1=a_1<a_2<\cdots$ be the sequence of integers defined by $a_1=1$ and $a_{k+1}$ is the smallest integer $n$ for which the number of solutions to $a_i+a_j \leq n$ (with $i\leq j\leq k$) is less than $n-k$.
Is the number of solutions to $a_i+a_j \leq x$ equal to $x+O(x^{1/4+o(1)})$?
This sequence was constructed by Rosen. Note that the number of solutions to $a_i+a_j\leq x$ is always at least $x$ by construction. Erdős and Rosen could not even prove whether the number of solutions to $a_i+a_j\leq x$ satisfies is at most $(1+o(1))x$.
The sequence begins
\[1,3,5,9,13,17,24,31,38,45,\ldots.\]
