{"schema":"vela.problem-packet.v0.1","problem":261,"statement":"Are there infinitely many $n$ such that there exists some $t\\geq 2$ and distinct integers $a_1,\\ldots,a_t\\geq 1$ 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 $2^{\\aleph_0}$ solutions?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}