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Exact density functionals for the exchange and correlation energies are\napproximated in practical calculations for the ground-state electronic\nstructure of a many-electron system. An important exact constraint for the\nconstruction of approximations is to recover the correct non-relativistic\nlarge-$Z$ expansions for the corresponding energies of neutral atoms with\natomic number $Z$ and electron number $N=Z$, which are correct to leading order\n($-0.221 Z^{5/3}$ and $-0.021 Z \\ln Z$ respectively) even in the lowest-rung or\nlocal density approximation. We find that hydrogenic densities lead to\n$E_x(N,Z) \\approx -0.354 N^{2/3} Z$ (as known before only for $Z \\gg N \\gg 1$)\nand $E_c \\approx -0.02 N \\ln N$. These asymptotic estimates are most correct\nfor atomic ions with large $N$ and $Z \\gg N$, but we find that they are\nqualitatively and semi-quantitatively correct even for small $N$ and for $N\n\\approx Z$. The large-$N$ asymptotic behavior of the energy is pre-figured in\nsmall-$N$ atoms and atomic ions, supporting the argument that widely-predictive\napproximate density functionals should be designed to recover the correct\nasymptotics. It is shown that the exact Kohn-Sham correlation energy, when\ncalculated from the pure ground-state wavefunction, should have no contribution\nproportional to $Z$ in the $Z\\to \\infty$ limit for any fixed $N$.\n