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
\[\lim \frac{1}{\log x}\sum_{a_n<x}\frac{1}{a_n}=0?\]
Erdős writes that it is easy to see that $\liminf a_n/n<\infty$ is possible, and that one can have
\[\sum_{a_n< x}\frac{1}{a_n}\gg \log\log x.\]
The upper density of such a sequence can be $1/2$, but probably not $>1/2$.
See also [359] and [867].
