I found a class of equations with the following form. $$A (Bm)^k | (Cm^2 + Dm + E)^n$$
$ m \ge 12$ can be any rational number, $n > k$ are natural numbers.
$ 0 < A < 1$ is fixed and the rest of the constants are fixedintegers A*(Bm)^k needs to be an integer.
My intuition says that there should be none but I don't know how to prove it either way.
Edit: Some more explanation of my intuition. I'd think that the right would increase too fast to be smaller than the right.