Nuclear mass contains a wealth of nuclear structure information, and has been widely employed to extract the nuclear effective interactions. The known nuclear mass is usually extracted from the experimental atomic mass by subtracting the masses of electrons and adding the binding energy of electrons in the atom. However, the binding energies of electrons are sometimes neglected in extracting the known nuclear masses. The influence of binding energies of electrons on nuclear mass predictions are carefully investigated in this work. If the binding energies of electrons are directly subtracted from the theoretical mass predictions, the rms deviations of nuclear mass predictions with respect to the known data are increased by about $200$ keV for nuclei with $Z, Ngeqslant 8$. Furthermore, by using the Coulomb energies between protons to absorb the binding energies of electrons, their influence on the rms deviations is significantly reduced to only about $10$ keV for nuclei with $Z, Ngeqslant 8$. However, the binding energies of electrons are still important for the heavy nuclei, about $150$ keV for nuclei around $Z=100$ and up to about $500$ keV for nuclei around $Z=120$. Therefore, it is necessary to consider the binding energies of electrons to reliably predict the masses of heavy nuclei at an accuracy of hundreds of keV.
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