Measurement of the Inclusive Cross Sections of Prompt $J\mathrm{/}\psi$ and $\psi (3686)$ Production in $e^{+}{e}^{-}$ Annihilation from $\sqrt{s}=3.808$ to $4.951$ GeV

The inclusive cross sections of prompt $J\mathrm{/}\psi$ and $\psi (3686)$ production are measured at center-of-mass energies from 3.808 to 4.951 GeV. The dataset used is 22 fb${}^{-1}$ of $e^{+}{e}^{-}$ annihilation data collected with the BESIII detector operating at the BEPCII storage ring. The results obtained are in agreement with the previous BESIII measurements of exclusive $J\mathrm{/}\psi$ and $\psi (3686)$ production. The average values obtained for the cross sections measured in the center-of-mass energy ranges from 4.527 to 4.951 GeV for $J\mathrm{/}\psi$ and from 4.843 to 4.951 GeV for $\psi (3686)$, where the impact of known resonances is negligible, are $14.0\pm 1.7\pm 3.1$ pb and $15.3\pm 3.0$ pb, respectively. For $J\mathrm{/}\psi$, the first and the second uncertainties are statistical and systematic, respectively. For $\psi (3686)$, the uncertainty is total. These values are useful for testing charmonium production models.