Publications de l'Institut Mathematique 2010 Volume 87, Issue 101, Pages: 121-128
doi:10.2298/PIM1001121D
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On the coprimality of some arithmetic functions

De Koninck Jean-Marie, Kátai Imre

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

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

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