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]