Geodesic
On a sequence formed by iterating a divisor operator
Czechoslovak Mathematical Journal, Tome 69 (2019) no. 4, pp. 1177-1196 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $$ \underbrace {d^{s}( d^{s}( \ldots d^{s}( d^{s}( n) +\ell ) +\ell \ldots ) +\ell ) }_{m\text {-times}} =\begin {cases} d^{s}( n) \text {for} \ m=1,\\ d^{s}( d^{s}( n) +\ell ) \text {for} \ m=2,\\ d^{s}(d^{s}( d^{s}(n) +\ell ) +\ell ) \text {for} \ m=3, \\ \vdots \end {cases} $$ We present classical and nonclassical notes on the sequence $ ( \delta ^{s,\ell ,m}( n)) _{m\geq 1}$, where $\ell $, $n$, $s$ are understood as parameters.
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 $$ \underbrace {d^{s}( d^{s}( \ldots d^{s}( d^{s}( n) +\ell ) +\ell \ldots ) +\ell ) }_{m\text {-times}} =\begin {cases} d^{s}( n) \text {for} \ m=1,\\ d^{s}( d^{s}( n) +\ell ) \text {for} \ m=2,\\ d^{s}(d^{s}( d^{s}(n) +\ell ) +\ell ) \text {for} \ m=3, \\ \vdots \end {cases} $$ We present classical and nonclassical notes on the sequence $ ( \delta ^{s,\ell ,m}( n)) _{m\geq 1}$, where $\ell $, $n$, $s$ are understood as parameters.
DOI : 10.21136/CMJ.2019.0133-18
Classification : 03H05, 11A25, 11A41
Keywords: divisor function; prime number; iterated sequence; internal set theory 
@article{10_21136_CMJ_2019_0133_18,
     author = {Djamel, Bellaouar and Abdelmadjid, Boudaoud and \"Ozer, \"Ozen},
     title = {On a sequence formed by iterating a divisor operator},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1177--1196},
     year = {2019},
     volume = {69},
     number = {4},
     doi = {10.21136/CMJ.2019.0133-18},
     mrnumber = {4039629},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2019.0133-18/}
}
TY  - JOUR
AU  - Djamel, Bellaouar
AU  - Abdelmadjid, Boudaoud
AU  - Özer, Özen
TI  - On a sequence formed by iterating a divisor operator
JO  - Czechoslovak Mathematical Journal
PY  - 2019
SP  - 1177
EP  - 1196
VL  - 69
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2019.0133-18/
DO  - 10.21136/CMJ.2019.0133-18
LA  - en
ID  - 10_21136_CMJ_2019_0133_18
ER  - 
%0 Journal Article
%A Djamel, Bellaouar
%A Abdelmadjid, Boudaoud
%A Özer, Özen
%T On a sequence formed by iterating a divisor operator
%J Czechoslovak Mathematical Journal
%D 2019
%P 1177-1196
%V 69
%N 4
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2019.0133-18/
%R 10.21136/CMJ.2019.0133-18
%G en
%F 10_21136_CMJ_2019_0133_18
Djamel, Bellaouar; Abdelmadjid, Boudaoud; Özer, Özen. On a sequence formed by iterating a divisor operator. Czechoslovak Mathematical Journal, Tome 69 (2019) no. 4, pp. 1177-1196. doi: 10.21136/CMJ.2019.0133-18

[1] Bellaouar, D.: Notes on certain arithmetic inequalities involving two consecutive primes. Malays. J. Math. Sci. 10 (2016), 253-268. | MR

[2] Bellaouar, D., Boudaoud, A.: Non-classical study on the simultaneous rational approximation. Malays. J. Math. Sci. 9 (2015), 209-225. | MR

[3] Boudaoud, A.: La conjecture de Dickson et classes particulière d'entiers. Ann. Math. Blaise Pascal 13 (2006), 103-109 French. | DOI | MR | JFM

[4] Boudaoud, A.: Decomposition of terms in Lucas sequences. J. Log. Anal. 1 (2009), Article 4, 23 pages. | DOI | MR | JFM

[5] Koninck, J.-M. De, Mercier, A.: 1001 problems in classical number theory. Ellipses, Paris (2004), French. | MR | JFM

[6] Diener, F., (eds.), M. Diener: Nonstandard Analysis in Practice. Universitext, Springer, Berlin (1995). | DOI | MR | JFM

[7] Diener, F., Reeb, G.: Analyse Non Standard. Enseignement des Sciences 40, Hermann, Paris (1989), French. | MR | JFM

[8] Erdős, P., Kátai, I.: On the growth of $ d_{k}( n) $. Fibonacci Q. 7 (1969), 267-274. | MR | JFM

[9] Jin, R.: Inverse problem for upper asymptotic density. Trans. Am. Math. Soc. 355 (2003), 57-78. | DOI | MR | JFM

[10] Kanovei, V., Reeken, M.: Nonstandard Analysis, Axiomatically. Springer Monographs in Mathematics, Springer, Berlin (2004). | DOI | MR | JFM

[11] Nathanson, M. B.: Elementary Methods in Number Theory. Graduate Texts in Mathematics 195, Springer, New York (2000). | DOI | MR | JFM

[12] Nelson, E.: Internal set theory: A new approach to nonstandard analysis. Bull. Am. Math. Soc. 83 (1977), 1165-1198. | DOI | MR | JFM

[13] Ramanujan, S.: Highly composite numbers. Lond. M. S. Proc. (2) 14 (1915), 347-409 \99999JFM99999 45.1248.01. | DOI | MR

[14] Robinson, A.: Non-standard Analysis. Princeton Landmarks in Mathematics, Princeton University Press, Princeton (1974). | MR | JFM

[15] Berg, I. P. Van den: Extended use of IST. Ann. Pure Appl. Logic 58 (1992), 73-92. | DOI | MR | JFM

[16] Berg, I. P. Van den, (eds.), V. Neves: The Strength of Nonstandard Analysis. Springer, Wien (2007). | DOI | MR | JFM

[17] Wells, D.: Prime Numbers: The Most Mysterious Figures in Math. Wiley, Hoboken (2005).

[18] Yan, S. Y.: Number Theory for Computing. Springer, Berlin (2002). | DOI | MR | JFM

Cité par Sources :