Let $k\ge 2$. 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 $D^{\prime }\subseteq D$ such that ${\sum }_{d\in D^{\prime }}\frac{1}{d}=1$?