Erdős [Er70] notes that \[f_k(N) \ll \frac{\log N}{\log\log N}.\] Indeed, let $A$ be such a set. This in particular implies that, for every $t$, there are $<k$ solutions to $t=ap$ with $a\in A$ and $p$ prime, whence \[\sum_{n\in A}\frac{1}{n}\sum_{p<N}\frac{1}{p}< k \sum_{t<N^2}\frac{1}{t} \ll \log N,\] and the bound follows since $\sum_{p<N}\frac{1}{p}\gg \log\log N$.

The analogous question with natural density in place of logarithmic density (that is, we measure $\lvert A\rvert$ in place of $\sum_{n\in A}\frac{1}{n}$) is the subject of [536]. In particular Erdős [Er70] has constructed $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert \gg N$ where no four have the same pairwise least common multiple, and hence the interest of the natural density problem is the $k=3$ case.

A related combinatorial problem is asked at [857].