I'm not sure about this, but I do have an idea about how you could approach this. Maybe you can prove that there is a subset you can make such that all numbers are "out of reach". For example, for k = 3 and the set is all numbers up until 27, the numbers 1, 4, 9 and 27 satisfy S.
Maybe if you order them and pick the th, th, th, th, etc. ?