  The odd-even staggering of neighboring nuclear masses is very useful in
calculating local mass relations and nucleon-pair correlations. During the past
decades, there has been an increasing interest in the odd-even features of the
mass relations and related quantities exhibited in masses of neighboring
nuclei. In this work, after choosing a nucleus, we made an analysis of its
neighboring nuclei on the upper left corner and the lower right corner
respectively. We empirically obtained a new residual interaction formula of
even-$A$ ($A$ is the mass number) nuclei, and it is an revision based on the
existing empirical local formula of the proton-neutron interactions between the
last proton and the last neutron ($\delta V_{1p-1n}$). We then calculated the
even-$A$ nuclear masses. The differences between our calculated values and the
AME2012 database show that the root-mean-squared deviations (RMSD) are small
(for even-$A$ nuclei: $A$ $\geq$ 42, RMSD $\approx$ 161 keV; $A$ $\geq$ 100,
RMSD $\approx$ 125 keV), while for heavy nuclei, some of our calculated values
can reach an accuracy of a few tens of keV. With our residual interaction
formula including one parameter, we have successfully predicted some unknown
masses. Some of our predicted values compared well with the experimental values
(AME2016). In addition, the accuracy and simplicity of our predicted masses for
medium and heavy nuclei are comparable to those of the AME2012 (AME2016)
extrapolations.
