Number fluctuation and the fundamental theorem of arithmetic

We consider N bosons occupying a discrete set of single-particle quantum states in an isolated trap. Usually, for a given excitation energy, there are many combinations of exciting different number of particles from the ground state, resulting in a fluctuation of the ground state population. As a counterexample, we take the quantum spectrum to be logarithms of the prime number sequence, and using the fundamental theorem of arithmetic, find that the ground state fluctuation vanishes exactly for all excitations. The use of the canonical or grand canonical ensembles, on the other hand, gives a substantial number fluctuation for the ground state. This is an example of a system where canonical and grand canonical ensemble averagings are not valid because of the peculiar nature of the quantum spectrum.