Erdős Problem #954

declared status 'open'. Formalized: no. 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 $x+O(x^{1/4+o(1)})$? Current best: Note that the number of solutions to $a_i+a_j\leq x$ is always at least $x$ by construction. Erd\H{o}s and Rosen could not even prove whether the number of solutions to $a_i+a_j\leq x$ satisfies is at most $(1+o(1))x$. Prize: no. OEIS: A390642. Tags: number theory.

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