  We discuss numerical aspects of instantons in two- and three-dimensional
$\phi^4$ theories with an internal $O(N)$ symmetry group, the so-called
$N$-vector model. Combining asymptotic transseries expansions for large
argument with convergence acceleration techniques, we obtain high-precision
values for certain integrals of the instanton that naturally occur in loop
corrections around instanton configurations. Knowledge of these numerical
properties are necessary in order to evaluate corrections to the large-order
factorial growth of perturbation theory in $\phi^4$ theories. The results
contribute to the understanding of the mathematical structures underlying the
instanton configurations.
