On the Coprimality of Some Arithmetic Functions


Jean-Marie De Koninck, Imre Kátai




Let $\varphi$ stand for the Euler function. Given a positive integer $n$, let $\sigma(n)$ stand for the sum of the positive divisors of $n$ and let $\tau(n)$ be the number of divisors of $n$. We obtain an asymptotic estimate for the counting function of the set $\{n:\gcd(\varphi(n),\tau(n))=\gcd(\sigma(n),\tau(n))=1\}$. Moreover, setting $l(n):=\gcd(\tau(n),\tau(n+1))$, we provide an asymptotic estimate for the size of $\#\{n\leq x:l(n)=1\}$.