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](http://oeis.org/A033879) 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.



Any results on $\gcd(N^2, D(N^2))$ where $N^2$ is deficient and $D(N^2)$ is the deficiency of $N^2$?
It's an odd number between 1 and $N^2$, of course. It can actually be quite large: with $f(n)=\gcd(n^2,2n^2-\sigma(n^2)),$
$$
f(26334) = f(2 \cdot 3^2 \cdot 7 \cdot 11 \cdot 19) = 3^2 \cdot 7^2 \cdot 11^2 \cdot 19^2 = 19263321.
$$


Since $f(n)$ is odd, a somewhat tighter upper limit is $m^2$ where $m=n/2^\nu$ is the largest odd divisor of $n$. The above example has $f(n)=m^2/9,$ which shows that it would be hard to improve on this 'trivial' bound. On the other side, $f(n)=1$ infinitely often (at prime powers, for example).


I'm not sure what other kinds of results you would be interested in.


