On the $k$-fold iterate of the sum of divisors function
Colloquium Mathematicum, Tome 147 (2017) no. 2, pp. 247-255
Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences
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 $n$.
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
Affiliations des auteurs :
Jean-Marie De Koninck 1 ; Imre Kátai 2
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author = {Jean-Marie De Koninck and Imre K\'atai},
title = {On the $k$-fold iterate of the sum of divisors function},
journal = {Colloquium Mathematicum},
pages = {247--255},
year = {2017},
volume = {147},
number = {2},
doi = {10.4064/cm6880-6-2016},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm6880-6-2016/}
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TY - JOUR AU - Jean-Marie De Koninck AU - Imre Kátai TI - On the $k$-fold iterate of the sum of divisors function JO - Colloquium Mathematicum PY - 2017 SP - 247 EP - 255 VL - 147 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.4064/cm6880-6-2016/ DO - 10.4064/cm6880-6-2016 LA - en ID - 10_4064_cm6880_6_2016 ER -
Jean-Marie De Koninck; Imre Kátai. On the $k$-fold iterate of the sum of divisors function. Colloquium Mathematicum, Tome 147 (2017) no. 2, pp. 247-255. doi: 10.4064/cm6880-6-2016
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