**Tian-Xiao He and Peter J.-S. Shiue**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 30, 2024, Number 3, Pages 557–574

DOI: 10.7546/nntdm.2024.30.3.557-574

**Full paper (PDF, 253 Kb)**

## Details

### Authors and affiliations

Tian-Xiao He

*Department of Mathematics, Illinois Wesleyan University
Bloomington, Illinois 61702, United States*

Peter J.-S. Shiue

*Department of Mathematical Sciences, University of Nevada, Las Vegas
Vegas, Nevada, 89154-4020, United States*

### Abstract

We study the divisibility of the sums of the odd power of consecutive integers, and for odd integers and , by using the Girard–Waring identity. Faulhaber’s approach for the divisibilities is discussed. Some expressions of power sums in terms of Stirling numbers of the second kind are represented.

### Keywords

- Divisibility
- Sum of powers of consecutive integers
- Girard–Waring identity
- Recursive sequence
- Arithmetic series
- Faulhaber’s theorem

### 2020 Mathematics Subject Classification

- 05A15
- 11B99
- 11B83

### References

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### Manuscript history

- Received: 11 September 2024
- Accepted: 1 October 2024
- Online First: 3 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).

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## Cite this paper

He, T.-X., & Shiue, P. J.-S. (2024). Divisibility of the sums of the power of consecutive integers. *Notes on Number Theory and Discrete Mathematics*, 30(3), 557-574, DOI: 10.7546/nntdm.2024.30.3.557-574.