Let $A\subseteq \mathbb{N}$ be infinite and $d_A(n)$ count the number of $a\in A$ which divide $n$. Is it true that, for every $k$,$$\underset{x\to \infty }{\mathrm{limsup}}\frac{{\max }_{n<x}{d}_A(n)}{{\left({\sum }_{n\in A\cap [1,x)}\frac{1}{n}\right)}^k}=\infty ?$$