  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.
