For a natural number $n\geq 1$, let $PF(n)$ denote the number of prime factors (with multiplicity) of $n$. For example, since $48=2*2*2*2*3$, we have $PF(48)=5$.

For any natural number $N\geq 1$, define $$E(N)=\frac{\sum_{k=1}^N PF(k)}{N},$$ the expected value for the number of prime factors for an integer between $1$ and $N$.

I have computed, for example, that $E(2^{22})\approx 3.63,$ and the sequence generally seems to increase. (Of course, the sequence $E(N)$ is not monotonically increasing, because it decreases when $N$ is prime. But when I evaluate $E(2^m)$ for various $m$, I find it is monotonically increasing for $m\leq 22$.

${\bf Question}$: Does there exist an elementary function $g\colon\{\mathbb N}\rightarrow {\mathbb N}$ such that $$E(n)\in\Theta(g(n)).$$ Here we are using "big-$\Theta$" notation; one function is said to be "big-$\Theta$" of another function if the first is asymptotically bounded above and below by a nonzero constant multiple of the second. By "elementary function," I mean the standard definition; roughly the closure under sum, product, quotient, and composition of exponential, log, and polynomial functions.