OPEN

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 \[\lim \frac{1}{\log x}\sum_{a_n<x}\frac{1}{a_n}=0?\]

#839: [Er78f]