SUMMARY
For which values of nn is it possible to color the positive integers using precisely nn colors in such a way that for any aa, the numbers a,2a,…,naa,2a,\dots,na all receive different colors? The third-named author posed the question around 2008-2009. Particular cases appeared in the Hungarian high school journal KöMaL in April 2010, and the general version appeared in May 2010 on MathOverflow, posted by D. Pálvölgyi. The question remains open. We discuss the known partial results and investigate a series of related matters attempting to understand the structure of these nn-satisfactory colorings.Specifically, we show that there is an nn-satisfactory coloring whenever there is an abelian group operation ?\oplus on the set {1,2,…,n}\{1,2,\dots,n\} that is compatible with multiplication in the sense that whenever ii, jj and ijij are in {1,…,n}\{1,\dots,n\}, then ij=i?jij=i\oplus j. This includes in particular the cases where n+1n+1 is prime, or 2n+12n+1 is prime, or n=p2-pn=p^2-p for some prime pp, or there is a kk such that q=nk+1q=nk+1 is prime and 1k,…,nk1^k,\dots,n^k are all distinct modulo qq (in which case we call qq a strong representative of order nn). The colorings obtained by this process we call multiplicative. We also show that nonmultiplicative colorings exist for some values of nn.There is an nn-satisfactory coloring of Z+\mathbb Z^+ if and only if there is such a coloring of the set KnK_n of nn-smooth numbers. We identify all nn-satisfactory colorings for n?n\leqslant 5 and all multiplicative colorings for n\leqslant 8n\leqslant 8, and show that there are as many nonmultiplicative colorings of K_nK_n as there are real numbers for n=6n=6 and 8. We show that if nn admits a strong representative qq then it admits infinitely many and in fact the set of such qq has positive natural density in the set of all primes.We also show that the question of whether there is an nn-satisfactory coloring is equivalent to a problem about tilings, and use this to give a geometric characterization of multiplicative colorings.