The cross sections for the $Z=$ 10 -- 19 with isotopes $T_{z}=-3/2$ to $-5$ in the 140$A$ MeV $^{40}$Ca + $^{9}$Be projectile fragmentation reaction have been predicted. An empirical formula based on the correlation between the cross section and average binding energy of isotope has been adopted to predict the cross section... The binding energies in the AME16, WS4, and the theoretical prediction by the spherical relativistic continuum Hartree$-$Bogoliubov theory have been used. Meanwhile, the {\sc fracs} parametrization and the modified statistical abrasion-ablation model are also used to predict the cross sections for the proton-rich isotopes. The predicted cross sections for the $T_{z}=-$3 isotopes are close to $10^{-10}$ mb, which hopefully can be studied in experiment. In addition, based on the predicted cross sections, the $Z=$ 14 is suggested to be a new magic number in the light proton-rich nuclei with $T_{z} \leq-$3/2, of which the phenomenon is much more evident than it is from the average binding energy per nucleon. (read more)