Count k for consecutive-sum generator

An array generator service produces a consecutive-integers array starting at a positive integer k: [k, k+1, ..., k+m−1] for some m ≥ 1. The service returns such an array if its sum equals a given positive integer s. Given s, determine how many distinct k values admit at least one valid array. Provide an algorithm, prove correctness, and analyze time and space complexity. For example, when s = 10, valid k are 1 and 10.