  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 $Z\to \infty$ limit for any fixed $N$.
