**József Bukor, Ferdinánd Filip and János T. Tóth**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 30, 2024, Number 3, Pages 538–546

DOI: 10.7546/nntdm.2024.30.3.538-546

**Full paper (PDF, 201 Kb)**

## Details

### Authors and affiliations

József Bukor

*Department of Informatics, J. Selye University
945 01 Komárno, Slovakia*

Ferdinánd Filip

*Department of Mathematics, J. Selye University
945 01 Komárno, Slovakia*

János T. Tóth

*Department of Mathematics, J. Selye University
945 01 Komárno, Slovakia*

### Abstract

In this paper we study the properties of the unbounded sequence of positive reals having asymptotic distribution function of the form . As a consequence, we immediately get information on the asymptotic behavior of the power means of order of function values of some arithmetic functions, e.g., the first prime numbers or the values of the prime counting function.

### Keywords

- Block sequences
- Asymptotic distribution function
- Power mean

### 2020 Mathematics Subject Classification

- 11B05
- 11N37

### References

- Axler, C. (2018). On the arithmetic and geometric means of the first n prime numbers.
*Mediterranean Journal of Mathematics*, 15, Article ID 93. - Bukor, J., Filip, F., & Tóth, J. T. (2019). On properties derived from different types of asymptotic distribution functions of ratio sequences.
*Publicationes Mathematicae Debrecen*, 95(1–2), 219–230. - Filip, F., & Tóth, J. T. (2010). Characterization of asymptotic distribution functions of ratio block sequences.
*Periodica Mathematica Hungarica*, 60(2), 115–126. - Gerard, J., & Washington, L. C. (2018). Sums of powers of primes.
*The Ramanujan Journal*, 45, 171–180. - Jakimczuk, R. (2005). A note on sums of powers which have a fixed number of prime factors.
*Journal of Inequalities in Pure and Applied Mathematics*, 6(2), Article 31. - Jakimczuk, R. (2007). The ratio between the average factor in a product and the last factor.
*Mathematical Sciences*, 1(3), 53–62. - Jakimczuk, R. (2010). Functions of slow increase and integer sequences.
*Journal of Integer Sequences*, 13, Article 10.1.1. - Landau, E. (1909).
*Handbuch der Lehre von der Verteilung der Primzahlen*. Druck und Verlag von B. G. Teubner, Lepzig und Berlin. - Šalát , T., & Znám, S. (1968). On the sum of prime powers.
*Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica*, 21, 21–25. - Sándor, J. (2012). On certain bounds and limits for prime numbers.
*Notes on Number Theory and Discrete Mathematics*, 18(1), 1–5. - Sándor, J., & Verroken, A. (2011). On a limit involving the product of prime numbers.
*Notes on Number Theory and Discrete Mathematics*, 17(2), 1–3. - Strauch, O. (2015). Distribution functions of ratio sequences. An expository paper.
*Tatra Mountains Mathematical Publications*, 64, 133–185. - Strauch, O., & Porubský, Š. (2005).
*Distribution of Sequences: A Sampler*. Peter Lang, Frankfurt am Main. - Strauch, O., & Tóth, J. T. (2001). Distribution functions of ratio sequences.
*Publicationes Mathematicae Debrecen*, 58(4), 751–778.

### Manuscript history

- Received: 18 March 2024
- Revised: 26 September 2024
- Accepted: 1 October 2024
- Online First: 1 October 2024

### Copyright information

Ⓒ 2024 by the Authors.

This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

## Related papers

## Cite this paper

Bukor, J., Filip, F., & Tóth, J. T. (2024). On positive sequences of reals whose block sequence has an asymptotic distribution function. *Notes on Number Theory and Discrete Mathematics*, 30(3), 538-546, DOI: 10.7546/nntdm.2024.30.3.538-546.