Let $n$ be a positive integer. Find all sets $\{r_1,\cdots,r_n\} \subset \mathbb{Z}/(2^n-1)\mathbb{Z}$ such that $$\left\{ \sum_{j\in S} r_j \mid S\subset \{1,\cdots,n\}, S\ne \emptyset \right\} $$ forms a complete residue system in $\mathbb{Z}/(2^n-1)\mathbb{Z}$.
My progress: my guess is that the answer is all sets of the form $\{x,2x,\cdots,2^{n-1}x\}$ where $x$ is a unit in $\mathbb{Z}/(2^n-1)\mathbb{Z}$. Define $f(S)=\sum_{j\in S} r_j$ in $\mathbb{Z}/(2^n-1)\mathbb{Z}$, then there exists $T$ such that $f(T)=0$ in $\mathbb{Z}/(2^n-1)\mathbb{Z}$. By injectivity we can show $T=\{1,\cdots,n\}$
I have also tried the generating function $$P(x)=\prod_{j=1}^n (1+x^{r_j})$$ which is $1$ mod $\frac{x^{2^n-1}-1}{x-1}$.
Any ideas to this problem would be appreciated.