Is there an infinite hierarchy of sequences whose reciprocals diverge, starting with the natural numbers?

Question

It is well known that the sum of the reciprocals of the function $f_0(n)=n$ (the harmonic series) diverges:

$$\sum_{n=1}^\infty\,\frac{1}{n}=\infty$$

Similarly, the sum of the reciprocals of the prime numbers $f_1(n)=p_n$ diverges.

We also know that, since $p_n \sim n \ln n$,

$$\lim_{n \to \infty}\frac{f_0(n)}{f_1(n)} = 0$$

Is there an infinite hierarchy of sequences $f_k:\mathbb{N}\rightarrow \mathbb{N}$ such that for all $k\in \mathbb{N}$

1. $\sum_{n=1}^\infty\,\frac{1}{f_k(n)}$ diverges

2. $\lim_{n \to \infty}\frac{f_k(n)}{f_{k+1}(n)} = 0$?