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

Abstract Winkler has proved that, if n and m are positive integers with n ≤ m ≤ n 2 5 and m ≡ n (mod 2), then there exist positive integers {xi} such that Σxi = n and Σx12 = m. Extending work of Erdős, Purdy, and Hensley, we show that the best upper limit for m is n2 − 23/2n3/2 + O (n5/4). For k ≥ 2, we show that {Σ(kxi): xi ∈ N , Σxi = n} contains {0, 1, …, ap,k(n)}, where ap,k(n) = (kn){1 − k1 + 1/kn−1/k + O (n−2/k + 1/k2)}.