Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where the order of the summation is important. For example, $f(4) = 6$ because $4$ can be written as:
- $4$
- $2 + 2$
- $2 + 1 + 1$
- $1 + 2 + 1$
- $1 + 1 + 2$
- $1 + 1 + 1 + 1$

Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd.

Let's think step by step and output the final answer within \boxed{}.
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