Erdős Problem #45 — Monochromatic sums of distinct divisors under $k$-colourings

One last Ramsey type problem: Let $n_k$ be the smallest integer (if it exists) for which if we colour the proper divisors of $n_k$ by $k$ colours then $n_k$ will be a monochromatic sum of distinct divisors, namely a sum of distinct divisors in a colour class. I am sure that $n_k$ exists for every $k$ but I think it is not even known if $n_2$ exists. It would be of some interest to determine at least $n_2$.