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.



What is the growth rate for divisibility of integers
This Theorem 430, on page 355 of Hardy and Wright, that he "average order" of $\Omega(n)$ is $\log \log n.$ Then they point out, formula 22.10.2, that
$$ \sum\_{n \leq x} \; \Omega (n) \; = \; x \log \log x + B\_2 x + o(x) $$ and say how to find the constant $$B\_2 = B\_1 + \sum\_p \; \frac{1}{p(p-1)}. $$
Previously , $B\_1$ was given as the constant in Merten's Theorem, first 427:
$$ \sum\_{p \leq x} \; \frac{1}{p} \; = \; \log \log x + B\_1 + o(1), $$
then theorem 428, 
$$ B\_1 = \gamma + \sum \left\{ \log \left( 1 - \frac{1}{p} \right) + \frac{1}{p} \right\}$$ 
so that $B\_1 = 0.26149721\ldots$


