  We consider the second R\'enyi entropy $S^{(2)}$ in pure lattice gauge theory
with $SU(2)$, $SU(3)$ and $SU(4)$ gauge groups, which serves as a first
approximation for the entanglement entropy and the entropic $C$-function. We
compare the results for different gauge groups using scale setting via the
string tension. We confirm that at small distances $l$ our approximation for
the entropic $C$-function $C(l)$, calculated for the slab-shaped entangled
region of width $l$, scales as $N_c^2 - 1$ in accordance with its
interpretation in terms of free gluons. At larger distances $l$ $C(l)$ is found
to approach zero for $N_c = 3, 4$, somewhat more rapidly for $N_c = 4$ than for
$N_c = 3$. This finding supports the conjectured discontinuity of the entropic
$C$-function in the large-$N$ limit, which was found in the context of AdS/CFT
correspondence and which can be interpreted as transition between colorful
quarks and gluons at small distances and colorless confined states at long
distances. On the other hand, for $SU(2)$ gauge group the long-distance
behavior of the entropic $C$-function is inconclusive so far. There exists a
small region of lattice spacings yielding results consistent with $N_c=3,4$,
while results from other lattice spacings deviate without clear systematics. We
discuss several possible causes for discrepancies between our results and the
behavior of entanglement entropy in holographic models.
