Geodesic
On the Coprimality of Some Arithmetic Functions
Publications de l'Institut Mathématique, _N_S_87 (2010) no. 101, p. 121 .

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

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 $\#\{n\leq x:l(n)=1\}$.
Classification : 11A05 11A25 11N37
Keywords: Arithmetic functions, number of divisors, sum of divisors 
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     author = {Jean-Marie De Koninck and Imre K\'atai},
     title = {On the {Coprimality} of {Some} {Arithmetic} {Functions}},
     journal = {Publications de l'Institut Math\'ematique},
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     year = {2010},
     language = {en},
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Jean-Marie De Koninck; Imre Kátai. On the Coprimality of Some Arithmetic Functions. Publications de l'Institut Mathématique, _N_S_87 (2010) no. 101, p. 121 . http://geodesic.mathdoc.fr/item/PIM_2010_N_S_87_101_a8/