(1) $\pi(n)$, the number of primes at most $n$, is asymptotic to $n / \ln n$.

(2) In the Erdős-Rényi random graph model, $p = \ln n / n$ is a sharp threshold for the connectedness of the graph $G(n,p)$ on $n$ vertices with edge-probability $p$.

Is there any connection between these two, or is the ratio $n / \ln n$ natural enough to arise in several unrelated circumstances by happenstance?

(I ask as neither an expert in random graphs nor in number theory.)