OPEN

Is it true that, for all sufficiently large $k$, there exist finite intervals $I_1,\ldots,I_k\subset \mathbb{N}$ with $\lvert I_i\rvert \geq 2$ for $1\leq i\leq k$ such that \[1=\sum_{i=1}^k \sum_{n\in I_i}\frac{1}{n}?\]

#289: [ErGr80]

number theory, unit fractions