Czechoslovak Mathematical Journal, first online, pp. 1-20


On a sequence formed by iterating a divisor operator

Bellaouar Djamel, Boudaoud Abdelmadjid, Özen Özer

Received March 12, 2018.   Published online September 16, 2019.

Abstract:  Let $\mathbb{N}$ be the set of positive integers and let $s\in\mathbb{N}$. We denote by $d^s$ the arithmetic function given by $ d^s( n) =( d( n) ) ^s$, where $d(n)$ is the number of positive divisors of $n$. Moreover, for every $\ell,m\in\mathbb{N}$ we denote by $\delta^{s,\ell,m}( n) $ the sequence $\underbrace{d^s( d^s( \ldots d^s( d^s( n) +\ell) +\ell \ldots) +\ell)}_{m\text{-times}} = d^s( n) \text{for} m=1, d^s( d^s( n) +\ell) \text{for} m=2, d^s(d^s( d^s(n) +\ell) +\ell) \text{for} m=3, \dots $ We present classical and nonclassical notes on the sequence $ ( \delta^{s,\ell,m}( n)) _{m\geq1}$, where $\ell,n,s$ are understood as parameters.
Keywords:  divisor function; prime number; iterated sequence; internal set theory
Classification MSC:  11A25, 11A41, 03H05
DOI:  10.21136/CMJ.2019.0133-18

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Affiliations:   Bellaouar Djamel (corresponding author), Department of Mathematics, University 8 Mai 1945-Guelma, B.P 24000, Guelma, Algeria, e-mail: bellaouardj@yahoo.fr, bellaouar.djamel@univ-guelma.dz; Boudaoud Abdelmadjid, Laboratory of Pure and Applied Mathematics (LMPA), University Mohamed Boudiaf-M'sila, B.P 28000 M'sila, Algeria, e-mail: boudaoudab@yahoo.fr, abdelmadjid.boudaoud@univ-msila.dz; Özen Özer, Department of Mathematics, University of Kirklareli, 39000 Kirklareli, Turkey, e-mail: ozenozer39@gmail.com, ozenozer@klu.edu.tr


 
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