The type of inequality we will encounter in this adventure is of the form
\[ n \le A \lfloor \log_p^+ n \rfloor + B \]
where \(A,B\ge 0\) are constants, \(p>1\) is a prime number, and \(\log_p^+ n = \max(0,\log_p n)\) is the positive part of the base-\(p\) logarithm of \(n\). The goal is to obtain a bound on \(n\) in terms of \(A, B\) and \(p\).