  Using time-dependent density functional theory (tdDFT) with exchange kernels
we calculate and test imaginary frequency-dependent dipole polarizabilities for
all atoms and many ns in rows 1-6 of the periodic table. These are then
integrated over frequency to produce $C_6$ coefficients. Results are presented
under different models: straight tdDFT calculations using two different
kernels, "benchmark" tdDFT calculations corrected by more accurate quantum
chemical and experimental data, and "benchmark" tdDFT with frozen orbital
anions. Parametrisations are presented for 411+ atoms and ions, allowing
results to be easily used by other researchers. A curious relationship,
$C_{6,XY}\propto [\alpha_X(0)\alpha_Y(0)]^{0.73}$ is found between $C_6$
coefficients and static polarizabilities $\alpha(0)$. The relationship
$C_{6,XY}=2C_{6,X}C_{6,Y}/[\alpha_X/\alpha_YC_{6,Y}+\alpha_Y/\alpha_XC_{6,X}]$
is tested and found to work well ($<5$\% errors) in about 80\% of cases, but
can break down badly ($>30$\% errors) in a small fraction of cases.
