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We prove there exist infinitely many odd integers $n$ for which there existsa 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 differentapproaches and methods.

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