Partitioning {1,2,...n} into consecutive blocks with equal sums

Title says it all, really. Suppose I want to partition {1, 2, ... n} into consecutive blocks, i.e.,

{1, 2, 3, ... a}, {a+1, ... b}, ..., {z+1, ... n}

for k such (nonempty) sets, such that the sums of the elements in each set are equal. For which k do there exist positive integer n with such a partition? As a simple example, k=2 and n=3 offers {1,2} U {3}, each with sum 3.