Extremal problems for subset divisors

Let A be a set of n positive integers. We say that a subset B of A is a divisor of A, if the sum of the elements in B divides the sum of the elements in A. We are interested in the following extremal problem. For each n, what is the maximum number of divisors a set of n positive integers can have? We determine this function exactly for all values of n. Moreover, for each n we characterize all sets that achieve the maximum. We also prove results for the k-subset analogue of our problem. For this variant, we determine the function exactly in the special case that n = 2k. We also characterize all sets that achieve this bound when n = 2k.