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