The sum of the squares of the parts of a partition, and some related questions

Winkler has proved that, if n and m are positive integers with n ≤ m ≤ n² 5 and m ≡ n (mod 2), then there exist positive integers {x_i} such that Σx_i = n and Σx₁² = m. Extending work of Erdo{combining double acute accent}s, Purdy, and Hensley, we show that the best upper limit for m is n² - 2^3/2n^3/2 + O(n^5/4). For k ≥ 2, we show that {Σ(_k^x_i): x_i ∈ N, Σx_i = n} contains {0, 1, ..., a_p,k(n)}, where a_p,k(n) = (_kⁿ){1 - k^{1 + 1/k}n^-1/k + O(n^{-2/k + 1/k2})}.