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

It is easy to see that $\liminf a_n/n<\infty$ is possible.

See also [359].