On a Diophantine equation of Erd\H{o}s and Graham

We study solvability of the Diophantine equation \begin{equation*} \frac{n}{2^{n}}=\sum_{i=1}^{k}\frac{a_{i}}{2^{a_{i}}}, \end{equation*} in integers $n,k,a_1,\dots ,a_k$ satisfying the conditions $k\ge 2$ and $a_i<a_{i+1}$ for $i=1,\dots ,k-1$. The above Diophantine equation (of polynomial-exponential type) was mentioned in the monograph of Erd\H{o}s and Graham, where several questions were stated. Some of these questions were already answered by Borwein and Loring. We extend their work and investigate other aspects of Erd\H{o}s and Graham equation. First of all, we obtain the upper bound for the value $a_k$ given in terms of $k$ only. This mean, that with fixed $k$ our equation has only finitely many solutions in $n, a_{1},\ldots, a_{k}$\mathrm{.}Moreover,weconstructaninfiniteset$\cal{K}$, such that for each $k\in \mathcal{K}$, the considered equation has at least five solutions. As an application of our findings we enumerate all solutions of the equation for $k\le 8$. Moreover, by applying greedy algorithm, we extend Borwein and Loring calculations and check that for each $n\le {10}^4$ there is a value of $k$ such that the considered equation has a solution in integers $n+1=a_1<a_2<\dots <a_k$. Based on our numerical calculations we formulate some further questions and conjectures.