Discret. Math. | 2019
Generating infinitely many satisfactory colorings of positive integers
Abstract
Abstract An n -satisfactory coloring of the n -smooth integers is an assignment of n colors to the positive integers whose prime factors are at most n so that for each such m , the integers m , 2 m , … , n m receive different colors. In this note, we give a short proof that infinitely many 6-satisfactory colorings of the 6-smooth integers exist and show how the technique of the proof can be applied more generally, including for n ∈ { 8 , 10 , 12 } .