Smallest Set of Weights for One-Pan Balance

Classic Problem
Consider a set of balance scales for determining the weight of a physical object.

Let this set of scales be such that weights may be placed in one of the pans.

What is the smallest set of weights needed to weigh any given integer weight up to a given amount?

Solution
A set of $m$ weights in the sequence $\sequence {2^n}$:
 * $1, 2, 4, 8, 16, \ldots$

allows one to weigh any given integer weight up to $2^m - 1$.

Also see

 * Smallest Set of Weights for Two-Pan Balance