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All Random Solved Random Open
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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].
Additional thanks to: Zachary Chase