OPEN

Are there infinitely many $n$ such that there exists some $t\geq 2$ and $a_1,\ldots,a_t\geq 1$ such that
\[\frac{n}{2^n}=\sum_{1\leq k\leq t}\frac{a_k}{2^{a_k}}?\]
Is this true for all $n$? Is there a rational $x$ such that
\[x = \sum_{k=1}^\infty \frac{a_k}{2^{a_k}}\]
has at least two solutions?

Related to [260].