OPEN
This is open, and cannot be resolved with a finite computation.
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 Erdős thought it probably could not be $>1/2$. In fact this is false - Freud
[Fr93] constructed a sequence with upper density $19/36$.
See also
[359] and
[867].
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Additional thanks to: Boris Alexeev
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #839, https://www.erdosproblems.com/839, accessed 2026-01-16
