Two divisors of $(n^2+1)/2$ summing up to $\delta n + \delta \pm 2$, $\delta$ even

Abstract

We prove there exist infinitely many odd integers $n$ for which there exists a pair of positive divisors $d_1, d_2>1$ of $(n^2+1)/2$ such that $$d_1+d_2=\delta n+(\delta+2).$$ We prove the similar result for $\varepsilon=\delta-2$ and $\delta\equiv4, 6\pmod{8}$ using different approaches and methods.