The production fraction of the $B_c^-$ meson with respect to the sum of $B^-$ and $\bar{B}^0$ mesons is measured in both 7 and 13 TeV center-of-mass energy $pp$ collisions produced by the Large Hadron Collider (LHC), using the LHCb detector. The rate, approximately 3.7 per mille, does not change with energy, but shows a transverse momentum dependence. The $B_c^- - B_c^+$ production asymmetry is also measured, and is consistent with zero within the determined statistical and systematic uncertainties of a few percent.