EMIS ELibM Electronic Journals Publications de l'Institut Mathématique, Nouvelle Série
Vol. 87(101), pp. 121–128 (2010)

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Jean-Marie De Koninck and Imre Katai

Dép. de mathématiques et de statistique, Université Laval, Québec, Québec G1V 0A6, Canada and Computer Algebra Department, Eötvös Lorand University, 1117 Budapest, Pazmany Péter Sétany I/C, Hungary

Abstract: 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 $#\{n\leq x:l(n)=1\}$.

Keywords: Arithmetic functions, number of divisors, sum of divisors

Classification (MSC2000): 11A05, 11A25, 11N37

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Electronic fulltext finalized on: 20 Apr 2010. This page was last modified: 18 Jan 2016.

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