Convergence-test using density exists?

Question

Let $0<a_n$. I want to declare that $\sum 1/a_n$ is convergent (or divergent) based on the behaviour of the density $S_n/n$, as $n\to\infty$, where $S_n$ is the number of elements of $a_n$ falling into $(0,n)$. For example, if $a_n$ is the n-th prime, then $S_n/n$ is about $1/\ln n$, and we know that the sum is divergent. Is the opposite true? I.e., from $S_n/n$ is about $1/\ln n$ does it follow that $\sum 1/a_n$ is convergent? Yes or no?