  We study the moduli dependence of a class of couplings in $K3\times T^2$
compactifications of type I string theory, for which one-loop amplitudes can be
written in terms of an N=2 supersymmetric index. This index is determined for
generic models as a function of the BPS spectrum. As an application we compute
the one-loop moduli dependence of the $F_g W^{2g}$ couplings, where W is the
N=2 gravitational superfield, for type I compactifications based on the
Gimon-Johnson K3 orientifolds, showing explicitly their dependence on the
aforementioned index.
