Let $A=\{n_1<n_2<\cdots \}$ be an infinite sequence of integers, and let ${\varphi }_A(k)$ count the number of $1\le m\le n_k$ such that the fraction $\frac{m}{n_k}$ does not have denominator $n_j$ for $j<k$ when written in lowest form; equivalently,$$\frac{n_k}{(m,n_k)}\ne n_j$$for all $1\le j<k$.Is there a sequence $A$ such that$$\underset{N\to \infty }{\lim }\frac{1}{N}\sum _{k\le N}\frac{{\varphi }_A(k)}{n_k}=0?$$