What an interesting question.
I don't know the answer but I have found the following links that may be relevant
First the question is why does the nucleus hold together at all when the electrostatic or coulomb force between the protons must be forcing the nucleus apart? The answer is the (strong) nuclear force described here:
http://en.wikipedia.org/wiki/Strong_interactionIn summary there was postulated a force from the interaction of neutrons and protons that held nuclei together. Now , the force is thought to be that that hold the proton and neutron particles
themselves together and the nuclear force is a by-product. So we have reached a point where we have an explanation why some nuclei are stable. The we need to think why some nuclei are stable and some unstable.
We move on to this article
http://en.wikipedia.org/wiki/Atomic_nucleusThe wikipedia article suggests in section "Shell models and other quantum models" that the nucleus has energy shells or levels roughly equivalent to electron energy levels . That section contains the phrase
present nuclear theory does predict the "magic numbers" of filled nuclear shells for both protons and neutrons.which
I interpret to mean (I may be wrong) there is no simple model for predicting whether a particular combination of neutrons and protons results in a stable or unstable nucleus.
However the next couple of sentences are:
The closure of the stable shells predicts unusually stable configurations, analogous to the noble group of nearly-inert gases in chemistry. An example is the stability of the closed shell of 50 protons, which allows tin to have 10 stable isotopes, more than any other element.
So it seems there are empirical rules to identify very stable nuclei
Now look at the powerpoint presentation
http://www.physics.gatech.edu/frog/lectures/ModernPhysicsLectures/MP18Nuclei.pptSlide 13 shows the line of stability
Slide 15 and 18 suggest most stable nuclei have even-even numbers of protons and neutrons a few stable nuclei have odd-odd numbers of protons and neutrons
Slides 20 and 21 suggest there are energy levels in the nucleus occupied by nucleons and the pairing or non-pairing of nucleons has a bearing on the nucelus stability - but I can't quite interpret that yet
Any other contributions or corrections to my interpretation?
Clive