Problem 839

Erdős Problem #839: declared status 'open'. Formalized: no. Let $1\leq a_1<a_2<\cdots$ be a sequence of integers such that no $a_i$ is the sum of consecutive $a_j$ for $j<i$. Is it true that$$\limsup \frac{a_n}{n}=\infty?$$Or even$$\lim \frac{1}{\log x}\sum_{a_n<x}\frac{1}{a_n}=0?$$ Prize: no. Tags: number theory.