In our Universe the equation for gravity is (GMm)/rn where r is the radius and n is the number of dimensions minus 1. This equation produces stable orbits in 2d and 3d but in 4 or more dimensions it does not produce stable orbits or at least it would be extremely difficult to produce stable orbits using this equation in more than 4 dimensions.
I found that if the equation (GMmsin(alogbr))/rn is used for gravity with r as the number of dimensions and n as the number of dimensions minus one it produces stable orbits in higher dimensions. I'm not sure if it produces stable orbits in any number of dimensions but it produces stable orbits in 4 spatial dimensions. It produces cogwheel shaped orbits. Because this equation would mean that gravity is attractive at some distances and repulsive at others this only allows stable orbits at certain distances with orbits impossible at other distances do to the repulsiveness of gravity. One thing this has in common with the normal law of gravity is that it produces a self similarity between the small scale and the large scale.