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Claim. Sz. Tengely, M. Ulas and J. Zygadło, On a Diophantine equation of Erdős and Graham, J. Number Theory 217 (2020), 445--459 (library card), study the equation

$$\frac{n}{2^n}=\sum_{i=1}^{k}\frac{a_i}{2^{a_i}},\qquad k\ge2,\quad a_1<\cdots<a_k,$$

of Problem 261. Using a modified greedy algorithm they verify that it has a solution for every $n\le10^4$, extending the computations of Borwein and Loring; this is the result the site's remarks credit. Their Theorem 2.1 gives necessary conditions on a solution with $k$ terms: $n\le2^{k+1}-k-2$, $n+1\le a_1\le n+3$, $2^{a_k-a_{k-1}}$ divides $a_k$, and $a_i=n+i$ for $i\le j$ whenever $n\ge2^{j+1}-j$; the bound $a_k\le2^{k+2}+2k(\log_2k-1)-4$ then leaves, for each fixed $k$, finitely many effectively computable solutions, which the paper enumerates for $k\le8$. The paper conjectures $a_k\le2(n+k)$ and proves it for $n\ge2^k-k$, constructs an infinite set of $k$ for which the equation has at least five solutions, and constructs an infinite set of rationals each with at least nine representations as a sum of terms $a_i/2^{a_i}$.

Covers. The second question for $1\le n\le10^4$: every such $n$ has the property. Not covered: $n>10^4$, so the second question stays open; the first question, answered on Borwein and Loring's page; and the third question, on which the multiplicity results bear without settling it.

Depends on. Nothing in this wiki; the claim rests on the cited paper.

Acceptance. Refereed: Journal of Number Theory 217 (2020), 445--459, DOI 10.1016/j.jnt.2020.05.006, in the December 2020 issue (refereed); the arXiv preprint 2008.01501 was posted on 4 August 2020, which dates this page. The site's curator credits the verification in the problem's remarks, but the site labels the problem OPEN, so the remark is not acceptance of the problem and the page lists no reviewed evidence. The library holds no file of the paper; the results are recorded from its card, and the corpus records no check of the proofs or of the computation.