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On the $k$-fold iterate of the sum of divisors function

Volume 147 / 2017

Jean-Marie De Koninck, Imre Kátai Colloquium Mathematicum 147 (2017), 247-255 MSC: Primary 11N37. DOI: 10.4064/cm6880-6-2016 Published online: 13 January 2017

Abstract

Let $\gamma(n)$ stand for the product of the prime factors of $n$. The index of composition $\lambda(n)$ of an integer $n\ge 2$ is defined as $\lambda(n)=\log n / \!\log \gamma(n)$ with $\lambda(1)=1$. Given an arbitrary integer $k\ge 0$ and letting $\sigma_k(n)$ be the $k$-fold iterate of the sum of divisors function, we show that, given any real number $\varepsilon \gt 0$, $\lambda(\sigma_k(n)) \lt 1+\varepsilon$ for almost all positive integers $n$.

Authors

  • Jean-Marie De KoninckDépartement de mathématiques
    Université Laval
    Québec G1V 0A6, Canada
    e-mail
  • Imre KátaiComputer Algebra Department
    Eötvös Lorand University
    Pázmány Péter sétány 1/C
    1117 Budapest, Hungary
    e-mail

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