Any results on $\gcd(N^2, D(N^2))$ where $N^2$ is deficient and $$D(N^2)=2N^2 - \sigma(N^2)$$ is the deficiency of $N^2$?
I checked OEIS sequence A033879 and have so far been able to get hold of Klyve, et. al's paper titled "On the difference between an integer and the sum of its proper divisors". Google does not appear to be of much help either.