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We consider the second Rényi 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 -functioNc (l), calculated for the slab-shaped entangled region of width l, scales as Nc² - 1 in accordance with its interpretation in terms of free gluons. At larger distances l C(l) is found to approach zero for Nc = 3, 4, somewhat more rapidly for Nc = 4 than for Nc = 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 Nc = 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. [ABSTRACT FROM AUTHOR]