  We study nuclear modification factors for single $D$ meson and semileptonic
decay lepton $l$ ($=e,\mu$) production in minimum bias proton-nucleus (p$A$)
collisions at the LHC in the color-glass-condensate (CGC) framework at leading
order in strong coupling. In our numerical computations, transverse momentum
($k_\perp$) dependent multi-point Wilson line correlators are employed for
describing target nucleus for p$A$ and proton for pp. The projectile proton is
treated with unintegrated gluon distribution function, which is also
$k_\perp$-dependent. The rapidity evolutions of these functions in the small
Bjorken $x$ region are taken into account by solving running coupling
Balitsky-Kovchegov (BK) equation at leading logarithmic accuracy. For
simplicity, we employ Kartvelishvili's type fragmentation function and a simple
model for lepton energy distribution from seileptonic decay, respectively, to
compute differential cross sections for $D$ and $l$ production. The gluon
saturation scale inside the heavy nucleus is enhanced and dependent on $x$,
which we take into account by replacing the initial saturation scale in the BK
equation with a larger value for the heavy nucleus. We show that the saturation
effect leads to perceptible nuclear suppression of $D$ production at forward
rapidity. Our numerical results predict similar nuclear suppressions in p$A$
collisions for forward $l$ production at lower transverse momentum
$p_\perp<2\;{\rm GeV}$. Numerical tables on the nuclear modifications of $D$
and $l$ are listed in this note.
