Let $A=\{n_1<n_2<\cdots\}$ be an infinite sequence of integers, and let $\phi_A(k)$ count the number of $1\leq m\leq n_k$ such that the fraction $\frac{m}{n_k}$ does not have denominator $n_j$ for $j<k$ when written in lowest form; equivalently, $$\frac{n_k}{(m,n_k)}\neq n_j$$ for all $1\leq j<k$.

Let $A=\{n_1<n_2<\cdots\}$ be an infinite sequence of integers, and let $\phi_A(k)$ count the number of $1\leq m\leq n_k$ such that the fraction $\frac{m}{n_k}$ does not have denominator $n_j$ for $j<k$ when written in lowest form; equivalently, $$\frac{n_k}{(m,n_k)}\neq n_j$$ for all $1\leq j<k$.

Is there a sequence $A$ such that $$\lim_{N\to \infty}\frac{1}{N}\sum_{k\leq N}\frac{\phi_A(k)}{n_k}=0?$$

This was solved by Haight [Ha] who proved that such a sequence does exist (contrary to Erdős' expectations).