What might the growth rate of
$$S(n)=\sum_{k=1}^{\lfloor n/2 \rfloor-1} \pi(n-2k)^{k+1}$$
(where $\pi$ is being used to represent the prime-counting function) be and what techniques are suggested to prove it?
Is the prime number theorem needed here?
(I am trying to use this function as a way to bound another one)