Integers with a sum of co-divisors yielding a square

Finding elliptic curves with high ranks has been the focus of much research. Recently, with the goal of generating elliptic curves with a large rank, some authors used large integers n which have many divisors, amongst which one can find divisors d such that \(d+n/d\) $d+n/d$ is a perfect square. This strategy is in itself a motivation for studying the function \(\tau _\Box (n)\) ${\tau }_{\square }(n)$ which counts the number of divisors d of an integer n for which \(d+n/d\) $d+n/d$ is a perfect square. We show that \(\sum _{n\le x} \tau _\Box (n) = c_\Box x^{3/4} +O(\sqrt{x})\) ${\sum }_{n\le x}{\tau }_{\square }(n)=c_{\square }{x}^{3/4}+O(\sqrt{x})$ for some explicit constant \(c_\Box \) $c_{\square }$ . Moreover, letting \(\rho _1(n):=\max \{d\mid n: d\le \sqrt{n}\}\) ${\rho }_1(n):=max\{d\mid n:d\le \sqrt{n}\}$ and \(\rho _2(n):=\min \{d\mid n: d\ge \sqrt{n}\}\) ${\rho }_2(n):=min\{d\mid n:d\ge \sqrt{n}\}$ stand for the middle divisors of n, we show that the order of magnitude of the number of positive integers \(n\le x\) $n\le x$ for which \(\rho _1(n)+\rho _2(n)\) ${\rho }_1(n)+{\rho }_2(n)$ is a perfect square is \(x^{3/4}/\log x\) $x^{3/4}/logx$ .