Counting integers of certain forms with applications to ternary quadratic forms

Abstract

For $\ell$ and $r$ fixed positive integers with $r<\ell$, we determine the number of positive integers $n \leq N$ which are of the forms $\ell^a(\ell b + r)$, $\ell^{2a}(\ell b + r)$, $\ell^{2a+1}(\ell b + r)$ for some nonnegative integers $a$ and $b$; and for fixed $r \in \{1,3,5,7\}$ we find the number of positive integers $n \leq N$ of each of the forms $2^{2a}(8 b + r)$, and $2^{2a+1}(8 b + r)$ for some nonnegative integers $a$ and $b$. Then we use our results to count the number of positive integers $n \leq N$ which can be represented by certain ternary quadratic forms.