If each integer has at most $r$ representations $m = pa$ with $p$ prime and $a \in A \subseteq [1, N]$, what is the best upper bound for $\sum_{a \in A} 1/a$? The candidate proof gives the matching order $\Theta_r(\log N / \log\log N)$.

Let $r\geq 2$ and suppose that $A\subseteq\{1,\ldots,N\}$ is such that, for any $m$, there are at most $r$ solutions to $m=pa$ where $p$ is prime and $a\in A$. Give the best possible upper bound for $\sum_{n\in A}\frac{1}{n}$.

The order is known — ∑ 1/n = Θ_r(log N / loglog N) (see erdos_538.matching_order) — but the sharp constant is not. This asks whether maxMass r N has a well-defined leading constant c_r in c_r · log N / loglog N.