  Finding minimum energy distribution of $N$ charges on a sphere is known as
the Thomson problem. Here, we study the vibrational properties of the $N$
charges in the lowest energy state within the harmonic approximation for $10\le
N\le 200$ and for selected sizes up to $N=372$. The maximum frequency
$\omega_{\rm max}$ increases with $N^{3/4}$, which is rationalized by studying
the lattice dynamics of a two-dimensional triangular lattice. The
$N$-dependence of $\omega_{\rm max}$ identifies magic numbers of $N=12, 32, 72,
132, 192, 212, 272, 282$, and 372, reflecting both a strong degeneracy of
one-particle energies and an icosahedral structure that the $N$ charges form.
$N=122$ is not identified as a magic number for $\omega_{\rm max}$ because the
former condition is not satisfied. The magic number concept can hold even when
an average of high frequencies is considered. The maximum frequency mode at the
magic numbers has no anomalously large oscillation amplitude (i.e., not a
defect mode).
