×

zbMATH — the first resource for mathematics

On a sequence formed by iterating a divisor operator. (English) Zbl 07144884
Summary: 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 \(l,m\in\mathbb{N}\) we denote by \(\delta^{s,l,m}(n)\) the sequence \[\underbrace{d^s(d^s(\ldots d^s(d^s(n)+l)+l\ldots)+l)}_{m\text{-times}}=\begin{cases}d^s(n)&\text{for }m=1,\\ d^s(d^s(n)+l)&\text{for }m=2,\\ d^s(d^s(d^s(n)+l)+l)&\text{for }m=3,\\ \vdots&\end{cases}\] We present classical and nonclassical notes on the sequence \((\delta^{s,l,m}(n))_{m\geq 1}\), where \(l\), \(n\), \(s\) are understood as parameters.
MSC:
11A25 Arithmetic functions; related numbers; inversion formulas
11A41 Primes
03H05 Nonstandard models in mathematics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bellaouar, D., Notes on certain arithmetic inequalities involving two consecutive primes, Malays. J. Math. Sci. 10 (2016), 253-268
[2] Bellaouar, D.; Boudaoud, A., Non-classical study on the simultaneous rational approximation, Malays. J. Math. Sci. 9 (2015), 209-225
[3] Boudaoud, A., La conjecture de Dickson et classes particulière d’entiers, Ann. Math. Blaise Pascal 13 (2006), 103-109 French
[4] Boudaoud, A., Decomposition of terms in Lucas sequences, J. Log. Anal. 1 (2009), Article 4, 23 pages
[5] Koninck, J.-M. De; Mercier, A., 1001 problems in classical number theory, Ellipses, Paris (2004), French
[6] Diener, F.; (eds.), M. Diener, Nonstandard Analysis in Practice, Universitext, Springer, Berlin (1995)
[7] Diener, F.; Reeb, G., Analyse Non Standard, Enseignement des Sciences 40, Hermann, Paris (1989), French
[8] Erdős, P.; Kátai, I., On the growth of \(d_k( n) \), Fibonacci Q. 7 (1969), 267-274
[9] Jin, R., Inverse problem for upper asymptotic density, Trans. Am. Math. Soc. 355 (2003), 57-78
[10] Kanovei, V.; Reeken, M., Nonstandard Analysis, Axiomatically, Springer Monographs in Mathematics, Springer, Berlin (2004)
[11] Nathanson, M. B., Elementary Methods in Number Theory, Graduate Texts in Mathematics 195, Springer, New York (2000)
[12] Nelson, E., Internal set theory: A new approach to nonstandard analysis, Bull. Am. Math. Soc. 83 (1977), 1165-1198
[13] Ramanujan, S., Highly composite numbers, Lond. M. S. Proc. (2) 14 (1915), 347-409 \99999JFM99999 45.1248.01
[14] Robinson, A., Non-standard Analysis, Princeton Landmarks in Mathematics, Princeton University Press, Princeton (1974)
[15] Berg, I. P. Van den, Extended use of IST, Ann. Pure Appl. Logic 58 (1992), 73-92
[16] Berg, I. P. Van den; (eds.), V. Neves, The Strength of Nonstandard Analysis, Springer, Wien (2007)
[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.