For $n \in \mathbb{N}$, consider non-negative valued functions $f$ on $\{1,2, \cdots , n\}$ satisfying $f(i) \geq f(j)$ for $i > j$ and $\sum_{i=1}^{n} (i + f(i)) = 2023.$ Choose $n$ such that $\sum_{i=1}^{n} f(i)$ is minimized. How many such functions exist in that case?

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