Let $N \geq 1$ and let $k (N)$ be maximal such that there are $k$ disjoint $A_{1}, \dots, A_{k} \subseteq {1, \dots, N}$ with $\sum_{n \in A_{i}} \frac{1}{n} = 1$ for all $i$ . Estimate $k (N)$ . Is it true that $k (N) = o (lo g N)$ ?

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number theory · solved · possible · 0 attempts

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