I encountered this identity while studying about the Kac determinants in CFT.
$$\sum_{\{n_1 + \cdot \cdot \cdot + n_k \}} k = \sum_{pq \le N} P(N-pq)$$
Here $P(N-pq)$ is the number of partitions of $N-pq$. $p,q$ are positive c-prime integers such that their product is less than equal to $N$, and $k$ is the number of parts in a partition of $N$. So $N=n_1 + n_2 + \cdot \cdot + n_k$ for some $k$ ($5=3+2 \implies k=2$), this sum is taken over all possible tuples.
How do I prove this identity?
I am a physics undergrad, so assumption of minimal prerequisites to this problem will be highly appreciated.