When you roll three octahedron (eight sided) dice with the sides numbered 1, 4, 16, 64, 256, 1024, 4096, 16384, there are 120 different sums which can be produced. The maximal value, in this example, is 16384.

Find the eight positive integers that could number a octahedral die and minimize the maximum value of the eight faces, while still resulting in 120 possible sums when three of the dice are rolled.