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?