  We investigate the impact of the neutron-star matter equation of state on the
$f$- and $p_1$-mode oscillations of neutron stars obtained within the Cowling
approximation and linearized general relativity. The $f$- and $p_1$-mode
oscillation frequencies, and their damping times are calculated using
representative sets of Skyrme Hartree-Fock and relativistic mean-field models,
all of which reproduce nuclear systematics and support $2M_\odot$ neutron
stars. Our study shows strong correlations between the frequencies of $f$- and
$p_1$-modes and their damping times with the pressure of $\beta$-equilibrated
matter at densities equal to or slightly higher than the nuclear saturation
density $\rho_0$. Such correlations are found to be almost independent of the
composition of the stars. The frequency of the $p_1$-mode of $1.4M_\odot$ star
is strongly correlated with the slope of the symmetry energy $L_0$ and
$\beta$-equilibrated pressure at density $\rho_0$. Compared to GR calculations,
the error in the Cowling approximation for the $f$-mode is about 30\% for
neutron stars of low mass, whereas it decreases with increasing mass. The
accuracy of the $p_1$-mode is better than 15\% for neutron stars of maximum
mass, and improves for lower masses and higher number of radial nodes.
