A set of integers is called a basis of order h if every nonnegative integer can be written as the sum of h not necessarily distinct elements of the set.(Nathanson, Additive Number Theory, page 7)

Thus, for example, the squares are a basis of order 4. (LaGrange's Theorem)

Consider the set of numbers S={ [n^(3/2)] | n an integer}, where the square brackets indicate the familiar floor function. This begins {0,1,2,5,8,11,14,18,22...}. S is certainly not a basis of order 2; there is for example no way to write 17 as a sum of two numbers from S. Is S a basis of order 3?

More generally, for alpha greater than 1 , let S={ [n^alpha] | n an integer}. For a given integer k is there a least alpha such that S is a basis of order k? If so, what is alpha?