Problem 783

Erdős Problem #783: declared status 'solved'. Formalized: no. Fix some constant $C>0$ and let $n$ be large. Let $A\subseteq \{2,\ldots,n\}$ be such that $(a,b)=1$ for all $a\neq b\in A$ and $\sum_{n\in A}\frac{1}{n}\leq C$. What choice of such an $A$ minimises the number of integers $m\leq n$ not divisible by any $a\in A$? Is this minimised by letting $n\geq q_1>q_2>\cdots$ be the consecutive primes in decreasing order and choosing $A=\{q_1,\ldots,q_k\}$ where $k$ is maximal such that$$\sum_{i=1}^k\frac{1}{q_i}\leq C?$$ Prize: no. Tags: number theory.