  In the context of f(R) gravity theories, we show that the apparent mass of a
neutron star as seen from an observer at infinity is numerically calculable but
requires careful matching, first at the star's edge, between interior and
exterior solutions, none of them being totally Schwarzschild-like but
presenting instead small oscillations of the curvature scalar R; and second at
large radii, where the Newtonian potential is used to identify the mass of the
neutron star. We find that for the same equation of state, this mass definition
is always larger than its general relativistic counterpart. We exemplify this
with quadratic $R^2$ and Hu-Sawicki-like modifications of the standard General
Relativity action. Therefore, the finding of two-solar mass neutron stars
basically imposes no constraint on stable f(R) theories. However, star radii
are in general smaller than in General Relativity, which can give an
observational handle on such classes of models at the astrophysical level. Both
larger masses and smaller matter radii are due to much of the apparent
effective energy residing in the outer metric for scalar-tensor theories.
Finally, because the f(R) neutron star masses can be much larger than General
Relativity counterparts, the total energy available for radiating gravitational
waves could be of order several solar masses, and thus a merger of these stars
constitutes an interesting wave source.
