  The high-energy atmospheric neutrino flux is dominated by neutrinos from the
decays of charmed hadrons produced in the forward direction by cosmic ray
interactions with air nuclei. We evaluate the charm contributions to the prompt
atmospheric neutrino flux as a function of the center-of-mass energy $\sqrt{s}$
of the hadronic collision and of the center-of-mass rapidity $y$ of the
produced charm hadron. Uncertainties associated with parton distribution
functions are also evaluated as a function of $y$. We find that the $y$
coverage of LHCb for forward heavy-flavour production, complemented by the
angular coverage of present and future forward neutrino experiments at the LHC,
bracket the most interesting $y$ regions for the prompt atmospheric neutrino
flux. At $\sqrt{s}=14$ TeV foreseen for the HL-LHC phase, nucleon collisions in
air contribute to the prompt neutrino flux prominently below $E_\nu\sim
10^7$~GeV. Measurements of forward charm and/or forward neutrinos produced in
hadron collisions up to $\sqrt{s}=100$ TeV, which might become possible at the
FCC, are relevant for the prompt atmospheric neutrino flux up to $E_\nu=10^8$
GeV and beyond.
