  We study the asymptotic behavior of the sequence $\{\Omega(n) \}_{ n \in
\mathbb{N} }$ from a dynamical point of view, where $\Omega(n)$ denotes the
number of prime factors of $n$ counted with multiplicity. First, we show that
for any non-atomic ergodic system $(X, \mathcal{B}, \mu, T)$, the operators
$T^{\Omega(n)}: \mathcal{B} \to L^1(\mu)$ have the strong sweeping-out
property. In particular, this implies that the Pointwise Ergodic Theorem does
not hold along $\Omega(n)$. Second, we show that the behaviors of $\Omega(n)$
captured by the Prime Number Theorem and Erd\H{o}s-Kac Theorem are disjoint, in
the sense that their dynamical correlations tend to zero.
