Question 199
What are the positive integers that can be obtained in exactly 2007 different ways, such as the sum of at least two consecutive positive integers? Which is the smallest of all of them? Example: The number $9$ is written in exactly two different ways: $$
\begin{aligned}
& 9=4+5 \\
& 9=2+3+4
\end{aligned}
$$
\begin{aligned}
& 9=4+5 \\
& 9=2+3+4
\end{aligned}
$$