[math-fun] Equal sums of powers

I recently saw in a book an example of two disjoint sets of integers whose sums of nth powers are equal for n = 1, 2, 3, 4, 5.* So given any integer M >= 5, I'm wondering whether there necessarily exist two disjoint sets of integers whose sums of nth powers are equal for all n in the range 1 <= n <= M. If this does not hold for all M >= 5, for which M does it hold? —Dan ———————————————————————————————————————————————————————————— * Given as {1, 6, 7, 17, 18, 23} and {2, 3, 11, 13, 21, 22}.