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 $S=\{ [n^{3/2}] \mid n\text{ an integer}\}$, where the square brackets indicate the familiar floor function. This begins $\{0,1,2,5,8,11,14,18,22,\ldots\}$. $S$ 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 $S(\alpha)=\{ [n^{\alpha}] \mid n\text{ an integer}\}$. For a given integer $k$ is there an $\alpha$ such that $S(\alpha)$ is a basis of order $k$? If so, what is the largest such $\alpha$?