Perturbative tests of non-perturbative counting

We observe that a class of quarter-BPS dyons in N=4 theories with charge vector (Q, P) and with nontrivial values of the arithmetic duality invariant I := gcd (Q wedge P) are nonperturbative in one frame but perturbative in another frame. This observation suggests a test of the recently computed nonperturbative partition functions for dyons with nontrivial values of the arithmetic invariant. For all values of I, we show that the nonperturbative counting yields vanishing indexed degeneracy for this class of states everywhere in the moduli space in precise agreement with the perturbative result.