Sum of all subsets of the set of even coprime integers relative to a power of a prime

by Ale Tolcachier   Last Updated June 30, 2020 02:20 AM

Let $p$ be an odd prime and $k\in \mathbb{N}$.

Let $S$ be the set of even coprime integers relative to $p^k$.

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 $k=1$ 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!

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