  Stimulated by the recent finding of Na$_{0.35}$CoO$_2$.1.3H$_2$O
superconductor, we investigate superconducting instabilities on a 2D triangular
lattice in the repulsive Hubbard model. Using the third-order perturbation
expansion with respect to the on-site repulsion $U$, we evaluate the linearized
Dyson-Gor'kov equation. We find that an $f$-wave spin-triplet pairing is the
most stable in a wide range of the next nearest neighbor hopping integral $t'$
and an electron number density $n$. The introduction of $t'$ is crucial to
adjust the van Hove singularities to the neighborhood of the Fermi surface
crossing around K point. In this case, the bare spin susceptibility shows the
broad peak around $\Gamma$ point. These conditions stabilize the $f$-wave
pairing. Although the $f$-wave pairing is also given by the
fluctuation-exchange approximation, the transition temperature is too low to be
observed. This is because the depairing effect by the spin fluctuation is
over-estimated. Thus, the third-order vertex corrections are important for the
spin-triplet superconductivity, like the case in Sr$_2$RuO$_4$.
