Let $p$ be an odd prime and $k\in \mathbb{N}$.
Let $S$ be the set of even coprime integers relative to $p^k$, i.e. $$S=\{2,4,\ldots,2\frac{p^k-1}{2}\}\setminus\{2p,\ldots,2p\frac{p^{k-1}-1}{2}\}.$$
For example, for $p=5, k=2$ then $$S=\{2,4,\ldots,24\}\setminus\{10,20\}=\{2,4,6,8,12,14,16,18,22,24\}$$
Question: If we take all the sums of all subsets of $S$ (with at least one element and without repeating elements), can we get all the even numbers up to the sum of all elements of $S$?
I could prove it easily by induction in the case of $k=1$ (in this case $S$ is simply $\{2,4,\ldots,p-1\}$) and I checked it for some low powers for example $9,25,27$. I think that this is true but I don't know how to manage a proof for $k\geq 2$ or find a counterexample. Maybe this is related to the Euler's totient function $\varphi$.
Any comment or sugerence will be appreciated. Thanks!