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

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!

• Can you give an example of $S$ for some $p^k$? I'm having trouble understanding its construction. – Carl Schildkraut Jun 30 at 2:50
• For example for 9, $S=\{2,4,8\}$, 2+4+8=14. In this case 6=2+4, 10=8+2, 12=4+8. – Ale Tolcachier Jun 30 at 2:52
• @AleTolcachier Is $p^k$ the upper bound for the elements in $S$ ? – Peter Jun 30 at 22:34
• If you mean the last element in $S$, then yes, the last element is $p^k-1$ so $p^k$ is "the" upper bound – Ale Tolcachier Jun 30 at 23:16