[math-fun] Division rings?

No, the title's been used already. So let's say `divisor necklaces' arising from the divisor chains which we were talking about a little while ago. Is it possible to arrange the numbers from 1 to n in a cycle, so that each is a divisor of the sum of the previous k members? There's a trivial example with n = 3 and k = 2, but I haven't found others. Is this because they don't exist, or is it just my incompetence? R.