  We study the effects of N=4 topological string amplitudes on the entropy of
black holes. We analyse the leading contribution associated to six-derivative
terms and find one particular operator which can correct the entropy of N=4
black holes. This operator is BPS-like and appears in the effective action of
type II string theory on K3 x T^2 or equivalently its heterotic dual on T^6. In
both descriptions the leading contribution arises at one-loop, which we
calculate explicitly on the heterotic side. We then consider whether this term
has any consequences for the entropy of (large) N=4 black holes and find that
it makes indeed a contribution at subleading order. Repeating the computation
for small black holes with vanishing horizon area at the classical level, we
prove that this coupling lifts certain flat directions in the entropy function
thereby being responsible for the attractor equations of some moduli fields.
