A complete comparison for the number of palindromes in different bases

Let $b$ and $b_1$ be distinct positive integers larger than $1$, and let $A_b(n)$ and $A_{b_1}(n)$ be the number of palindromes in bases $b$ and $b_1$ that are less than or equal to $n$, respectively. In this article, we finish the comparative study of the functions $A_b(n)$ and $A_{b_1}(n)$. As a result, we present the full picture of the asymptotic behavior of their difference.