Define the n-onion O_n as the union of all spheres S_k(n) about 0 in R^n whose radius is sqrt(k) for any integer k >= 0 (including the degenerate sphere S_0(n)). Then because of the sharpness of the 4-square theorem, O_1, O_2, and O_3 will not contain all the integer points in their respective spaces, while O_n for n >= 4 will. QUESTION: As long as we're discussing enumerating representations of 3-square integers, what about the 4-square ones (i.e., all integers) ? For a given k, how many points P(k) of the integer lattice in R^4 will have radius sqrt(k), i.e., lie on S_k(4) ??? --Dan