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

Jean-Marie De Koninck¹ ; Imre Kátai²

¹ Département de mathématiques Université Laval Québec G1V 0A6, Canada
² Computer Algebra Department Eötvös Lorand University Pázmány Péter sétány 1/C 1117 Budapest, Hungary

Colloquium Mathematicum, Tome 147 (2017) no. 2, pp. 247-255.

Résumé

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 $n$.

Détail

DOI : 10.4064/cm6880-6-2016

Keywords: gamma stand product prime factors index composition lambda integer defined lambda log log gamma lambda given arbitrary integer letting sigma k fold iterate sum divisors function given real number varepsilon lambda sigma varepsilon almost positive integers