Abstract

Let h, k be fixed positive integers, and let A be any set of positive integers. Let hA := {a_{1} + a_{2} + ⋯ + a_{r} : a_{i} ∈ A, r ≤ h} denote the set of all integers representable as a sum of no more than h elements of A, and let n(h, A) denote the largest integer n such that {1, 2, …, n} ⊆ hA. Let n(h, k) = max_{A} n(h, A), where the maximum is taken over all sets A with k elements. The purpose of this note is to determine n(h, A) when the elements of A are in arithmetic progression. In particular, we determine the value of n(h, 2).