Coloring the $n$-Smooth Numbers with $n$ Colors

For which values of $n$ is it possible to color the positive integers using precisely colors in such a way that for any $a$, numbers $a,2a,\dots,na$ all receive different colors? The third-named author posed question around 2008-2009. Particular cases appeared Hungarian high school journal KöMaL April 2010, and general version May 2010 on MathOverflow, posted by D. Pálvölgyi. remains open. We discuss known partial results investigate series related matters attempting understand structure these $n$-satisfactory colorings.
 Specifically, we show there an coloring whenever abelian group operation $\oplus$ set $\{1,2,\dots,n\}$ compatible with multiplication sense $i$, $j$ $ij$ are $\{1,\dots,n\}$, then $ij=i\oplus j$. This includes particular where $n+1$ prime, or $2n+1$ $n=p^2-p$ some prime $p$, $k$ $q=nk+1$ $1^k,\dots,n^k$ distinct modulo $q$ (in case call strong representative order $n$). colorings obtained this process multiplicative. also nonmultiplicative exist $n$.
 There $\mathbb Z^+$ if only $K_n$ $n$-smooth numbers. identify $n\leqslant 5$ multiplicative 8$, as many real $n=6$ 8. admits infinitely fact has natural density primes.
 whether equivalent problem about tilings, use give geometric characterization colorings.