An Asymptotic for Non-Wieferich Numbers

Abstract

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 $(n, (2^{\ell(n)}-1)/n)>1$, where $\ell(n)$ is the smallest positive integer with $2^{\ell(n)} \equiv 1 \mod n$. Franco and Pomerance showed that non-Wieferich numbers have density zero among the odd integers. We show that the counting function of $n \leq x$ in the symmetric difference of non-Wieferich numbers and cyclic numbers is $O(x/(\log_3(x))^2)$, under a mild assumption on the counting function of Wieferich primes.