According to Hardy-Ramanujan/Erdős-Kac we know that usually there are $\sim\log\log n$ prime numbers in a factorization. But if you pick up a natural number at random, and you factor it, what is the expected distribution of its factors?


By the Turán–Kubilius inquality, we see that this is the expectation:
$$ \left\{\frac{\log \log p}{\log \log n}:\ p\mid n\right\}$$
tend to equidistribute on $[0,1]$. 


Where can I find a more precise statement of the above? 


[Edit: a previous version contained comments irrelevant to the question]



$\log \log p / \log \log n$, where $p|n$, gets equidistributed in [0,1] (for almost all $n$)
For precise information on many questions of this type you should consult the Cambridge Tract, *Divisors* by Hall and Tenenbaum. Here is [Hildebrand's review](http://www.ams.org/journals/bull/1990-22-01/S0273-0979-1990-15871-9/S0273-0979-1990-15871-9.pdf) in the Bulletin of this book. A much more precise version of your question is a result of Erdős. Suppose $n =p\_1^{\alpha\_1} \cdots p\_k^{\alpha\_k}$ is the prime factorization of $n$ with $p\_1 < p\_2 < \ldots <p\_k$. Thus for almost all $n$, Hardy-Ramanujan tells us that $k=\omega(n) \sim \log \log n$. Let $\xi(n)$ be any function tending to infinity with $n$. For almost all $n$, Erdős showed that  

$$ 
|\log \log p\_i - i| \le (1+\epsilon) \sqrt{2i \log \log i}
$$ 
holds for all $\xi(n)\le i \le k$; see (3) of Hildebrand's review. There has been more recent progress in the field, but *Divisors* is still a good place to start. 


