Let $$D\_2(n) =\sum\_{pq\leq n} 1,$$
and 
$$F\_2(n) =\sum\_{pq\leq n} \frac{1}{pq}$$
where $p,q$ are primes. Similarly define
$$D\_k(n) =\sum\_{p\_1\cdots p\_k\leq n} 1,$$
and
$$F\_k(n) =\sum\_{p\_1\cdots p\_k\leq n} \frac{1}{p\_1\cdots p\_k}.$$


I believe my calculations showing
$$F\_2(n)\sim \frac{(\log\log n)^2}{2}$$
and
$$D\_2(n)\sim \frac{n \log\log n}{\log n}$$
are correct (I used PNT and Mertens theorems).


What about larger $k$? 


Is it possible to allow $k$ to grow very slowly and still get an estimate?



Estimating $\sum_{p_1\cdots p_k\leq n} \frac{1}{p_1\cdots p_k}$ for various $k$
For any fixed $k \geq 1$, one has $$D\_k(n) \sim \frac{x}{\log x} \frac{(\log \log x)^{k-1}}{(k-1)!}.$$ This was originally proved by Landau in 1909 using induction of $k$. 


For $k \ll \log \log x$, see Theorem 4 of chapter 6 of Tenenbaum's book "Introduction to analytic and probabilistic number theory." This is the Selberg-Delange method. 


One can get decent upper bounds uniformly in $k$ with elementary methods, see for instance page 4 of [these notes.](http://www.math.uiuc.edu/~ford/sieve_notes_shiftedprimes.pdf)


There is a nice heuristic: note the above assymptotic suggests a Poisson distribution with parameter $\log \log x$. One can view the events $\{ p | m\}\_{p}$ as weakly dependent events with probability of success about $1/p$. So by general principles of probability (section 4.7 of Sheldon Ross' book) suggest that the probability of $k$ successes is modeled by a Poisson distribution. The shift in the index $k$ comes from the fact that every integer has at least one prime factor. 


