On the $k$-fold iterate of the sum of divisors function
Volume 147 / 2017
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 $n$.
