Erdős problem 45

Let $k\geq 2$. Is there an integer $n_k$ such that, if $D=\{ 1<d<n_k : d\mid n_k\}$, then for any $k$-colouring of $D$ there is a monochromatic subset $D'\subseteq D$ such that $\sum_{d\in D'}\frac{1}{d}=1$?