Abdelkader Benyattou and Miloud Mihoubi

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 28, 2022, Number 2, Pages 234—239

DOI: 10.7546/nntdm.2022.28.2.234-239

**Download PDF (185 Kb)**

## Details

### Authors and affiliations

**Abdelkader Benyattou**

*Department of Mathematics and Informatics, Zian Achour University of Djelfa, Algeria
RECITS Laboratory, P. O. 32 Box 32, El Alia 16111, Algiers, Algeria*

**Miloud Mihoubi**

*Faculty of Mathematics, USTHB
RECITS Laboratory, P. O. 32 Box 32, El Alia 16111, Algiers, Algeria*

### Abstract

In this paper, we use the properties of the classical umbral calculus to determine sequences related to the Bell numbers and having periods divide .

### Keywords

- Classical umbral calculus
- Congruences
- Bell numbers

### 2020 Mathematics Subject Classification

- 05A40
- 11A07
- 11B73

### References

- Benyattou, A. (2020). Derangement polynomials with a complex variable.
*Notes on Number Theory and Discrete Mathematics*, 26(4), 128–135. - Benyattou, A., & Mihoubi, M. (2018). Congruences related to the Bell polynomials.
*Quaestiones Mathematicae*, 41(3), 437–448. - Benyattou, A., & Mihoubi, M. (2019). Real-rooted polynomials via generalized Bell umbra.
*Notes on Number Theory and Discrete Mathematics*, 25(2), 136–144. - Car, M., Gallardo, L. H., Rahavandrainy, O., & Vaserstein, L. N. (2008). About the period of Bell numbers modulo a prime.
*Bulletin of the Korean Mathematical Society*, 45(1), 143–155. - Gallardo, L. H. (2016). A property of the period of a Bell number modulo a prime number.
*Applied Mathematics E-Notes*, 16, 72–79. - Gertsch, A., & Robert, A. M. (1996). Some congruences concerning the Bell numbers.
*The Bulletin of the Belgian Mathematical Society – Simon Stevin*, 3, 467–475. - Gessel, I. M. (2003). Applications of the classical umbral calculus.
*Algebra Universalis*, 49, 397–434. - Maamra, M. S., & Mihoubi, M. (2014). The (
*r*_{1}, …,*r*)-Bell polynomials._{p}*Integers*, Article A34. - Mező, I. (2011). The
*r*-Bell numbers.*Journal of Integer Sequences*, (14), Article 11.1.1. - Mező, I., & Ramírez, J. L. (2017). Divisibility properties of the
*r*-Bell numbers and polynomials.*Journal of Number Theory*, 177, 136–152. - Montgomery, P. L., Nahm, J. R., & Wagstaff, S. (2010). The period of the Bell numbers modulo a prime. Mathematics of Computation, 79, 1793–1800.
- Radoux, C. (1975). Nombres de Bell modulo p premier et extensions de degré
*p*de*F*._{p}*Comptes Rendus de l’Academie des Sciences – Series A*, 281, 879–882. - Roman, S., & Rota, G. C. (1978). The umbral calculus. Advances in Mathematics, 27, 95–188.
- Rota, G. C. (1964). The Number of Partitions of a Set.
*The American Mathematical Monthly*, 71(5), 498–504. - Rota, G. C., & Taylor, B. D. (1994). The classical umbral calculus.
*SIAM Journal on Mathematical Analysis*, 25, 694–711. - Sun, Y., & Wu, X. (2011). The largest singletons of set partitions. European
*Journal of Combinatorics*, 32, 369–382. - Williams, G. T. (1945). Numbers generated by the function exp(
*e*− 1).^{x}*The American Mathematical Monthly*, 52, 323–327.

### Manuscript history

- Received: 7 June 2021
- Revised: 9 March 2022
- Accepted: 15 April 2022
- Online First: 19 April 2022

## Related papers

## Cite this paper

Benyattou, A., & Mihoubi, M. (2022). Note on some sequences having periods that divide (*p ^{p}* − 1) / (

*p*− 1).

*Notes on Number Theory and Discrete Mathematics*, 28(2), 234-239, DOI: 10.7546/nntdm.2022.28.2.234-239.