It is possible to write five as a sum in exactly six different ways:

$$\begin{align} &4 + 1\\ &3 + 2\\ &3 + 1 + 1\\ &2 + 2 + 1\\ &2 + 1 + 1 + 1\\ &1 + 1 + 1 + 1 + 1 \end{align}$$

How many different ways can one hundred be written as a sum of at least two positive integers?