OPEN
This is open, and cannot be resolved with a finite computation.
Is it true that, for all sufficiently large $k$, there exist finite intervals $I_1,\ldots,I_k\subset \mathbb{N}$, distinct, not overlapping or adjacent, with $\lvert I_i\rvert \geq 2$ for $1\leq i\leq k$ such that\[1=\sum_{i=1}^k \sum_{n\in I_i}\frac{1}{n}?\]
Erdős and Graham posed this in
[ErGr80] without the stipulation the intervals be distinct, non-overlapping, or adjacent, but Kovac in the comments has provided a simple argument showing that it is easily possible without this restriction, and likely
[ErGr80] just forgot to mention this natural restriction.
As an example representing $2$ rather than $1$, Hickerson and Montgomery, in the solution to AMS Monthly problem E2689 proposed by Hahn, found\[2=\sum_{i=1}^5 \sum_{n\in I_i}\frac{1}{n}\]where $I_1=[2,7]$, $I_2=[9,10]$, $I_3=[17,18]$, $I_4=[34,35]$, and $I_5=[84,85]$.
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This page was last edited 22 September 2025.
Additional thanks to: Vjekoslav Kovac
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 #289, https://www.erdosproblems.com/289, accessed 2026-01-16
