Nice problem. In what follows, we use the fact that for Harmonic Series $S_n = \sum_{1\le k\le n}\frac1k$, $S_n -\ln n \to \gamma$ where $\gamma$ is a positive constant. Consequently, $S_n\sim \ln n$ for $n$ large. Now, let $N$ be a large integer, and we count the number of pairs $(x,y)$ such that $f(x,y)=xy\le N$. Note that for $x=1$, $y$ can take precisely $N$ values, for $x=2$, $y$ can take $\lfloor N/2\rfloor$ values, and so on. Continuing, we find that the number of all such pairs is at least \[ \sum_{k=1}^N \left\lfloor \frac{N}{k}\right\rfloor \ge N\sum_{1\le k\le N}\frac1k - N = \Omega(N \ln N). \]That is, the number of pairs $(x,y)$ with $f(x,y)\le N$ is asymptotically at least $\Omega(N\log N)$, namely, one of the values in $\{1,2,\dots,N\}$ is attained at least $\Omega(\log N)$ times, so for $N$ large enough we can find $2023$ such pairs.