An Asymptotic for Non-Wieferich Numbers
A cyclic number is a positive integer $n$ such that $\gcd(n, \varphi(n)) = 1$. Erd\H{o}s showed that the number of cyclic numbers up to $x$, $C(x)$, has $C(x) \sim e^{-\gamma}x/\log_3(x)$ asymptotically as $x$ tends to infinity. We define Wieferich numbers to be odd, positive integers $n$ such that $(n, (2^{\ell(n)}-1)...