  A classic question in analytic number theory is to find asymptotics for
$\sigma_{k}(x)$ and $\pi_{k}(x)$, the number of integers $n\leq x$ with exactly
$k$ prime factors, where $\pi_{k}(x)$ has the added constraint that all the
factors are distinct. This problem was originally resolved by Landau in 1900,
and much work was subsequently done where $k$ is allowed to vary. In this paper
we look at a similar question about integers with a specific prime
factorization. Given $\boldsymbol{\alpha}\in\mathbb{N}^{k}$,
$\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2},...,\alpha_{k})$ let
$\sigma_{\boldsymbol{\alpha}}(x)$ denote the number of integers of the form
$n=p_{1}^{\alpha_{1}}... p_{k}^{\alpha_{k}}$ where the $p_{i}$ are not
necessarily distinct, and let $\pi_{\boldsymbol{\alpha}}(x)$ denote the same
counting function with the added condition that the factors are distinct. Our
main result is asymptotics for both of these functions.
