Problem 289

Erdős Problem #289: declared status 'open'. Formalized: yes. 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$$1=\sum_{i=1}^k \sum_{n\in I_i}\frac{1}{n}?$$ Prize: no. Tags: number theory, unit fractions.