Can we develop a formula for
$$ r(n)=\prod_{ \begin{array}{c} x,y\mid n \\ (x,y)=1 \end{array}} (x+y) $$
In words this is the product of sums of all coprime pairs of divisors of $n$.
For example $$r(12)=(1+2)(1+3)(1+4)(1+6)(1+12)(2+3)(3+4)=191100$$
There is some information on the number of coprime pairs of divisors (A063647).
In answer to the comment, this problem comes from wanting to know whether a certain function of $n$ divides at least one of the $(x+y)$ terms. This is true iff the function divides $r(n)$. I have no idea if a nice form exists, but I would certainly accept a form in terms of the prime factorization of $n$.