Let $\Omega(n)$ be the number of prime factors of $n$ with multiplicity, i.e., if $n=p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}$, $\Omega(n) = e_1 + e_2 + \cdots + e_k$ (OEIS). For example, for $n=9000 = 2^3 3^2 5^3$, $\Omega(n)=8$.
The graph below plots, for each $n$, the point $(n,\Omega(n))$, although on a log-scale; so I actually plot $q(n)= (\ln n,\Omega(n))$. Those are the red dots. The blue segments of the graph connect $q(n)$ to all those points below it on level $\Omega(n)-1$ that can reach $q(n)$ by multiplying by one prime. For example, $q(9000)$ would connect to $q(4500)$, $q(3000)$, and $q(1800)$, multiplying by $2$, $3$, and $5$ respectively. (The graph below only runs to $n=1024$.)
In contrast to
"[t]he chaotic course of $\Omega(n)$ through the natural numbers" (Wikipedia),
this graph is considerably more regular than I anticipated. Some aspects are easily explained. E.g., the parallel slanted lines represent powers of primes, with the leftmost slanted line passing through $q(2^m)$ up to $m=10$: $q(1024)= (6.93,10)$.
Could anyone explain the other regular qualitative features of this graph—the stairstep right end, the density patterns visible, the horizontal red-dot clustering, and any other discernible features?