Erdős Problem #1000 — Sequences with Vanishing Average Reduced-Denominator Count

For a strictly increasing sequence of positive integers $A=(n_k)_{k\geq0}$, define $\phi_A(k)$ to be the number of integers $m$ with $1\leq m\leq n_k$ such that

$$\frac{n_k}{\gcd(m,n_k)}\neq n_j$$

for every $j<k$. Equivalently, $m/n_k$ does not have denominator $n_j$ in lowest terms for any $j<k$. Define

$$\phi_{\mathrm{avg}}(N)=\frac1N\sum_{k=0}^{N-1}\frac{\phi_A(k)}{n_k}.$$

Does there exist a strictly increasing sequence of positive integers $A=(n_k)$ such that

$$\lim_{N\to\infty}\frac1N\sum_{k=0}^{N-1}\frac{\phi_A(k)}{n_k}=0?$$