Jose Arnaldo Bebita Dris and Immanuel Tobias San Diego

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 26, 2020, Number 2, Pages 27–33

DOI: 10.7546/nntdm.2020.26.2.27-33

**Full paper (PDF, 143 Kb)**

## Details

### Authors and affiliations

Jose Arnaldo Bebita Dris

*M. Sc. Graduate, Mathematics Department
De La Salle University, Manila, Philippines 1004
*

Immanuel Tobias San Diego

*Department of Mathematics and Physical Sciences
Trinity University of Asia, Quezon City, Philippines 1102
*

### Abstract

Let be an odd perfect number with special prime . In this article, we provide an alternative proof for the biconditional that holds if and only if . We then give an application of this result to the case when is a square.

### Keywords

- Sum of divisors
- Sum of aliquot divisors
- Deficiency
- Odd perfect number
- Special prime

### 2010 Mathematics Subject Classification

- 11A05
- 11A25

### References

- Broughan, K. A., Delbourgo, D., & Zhou, Q. (2013). Improving the Chen and Chen result for odd perfect numbers, Integers, 13, Article #A39.
- Chen, S.-C., & Luo, H. (2013). Odd multiperfect numbers, Bulletin of the Australian Mathematical Society, 88 (1), 56–63.
- Ewell, J. A. Jr. (1980). On the multiplicative structure of odd perfect numbers, Journal of Number Theory, 12, 339–342.
- Ochem, P. (2019). Answer to a question of the first author in Mathematics StackExchange, https://math.stackexchange.com/a/3151412/28816.
- Sloane, N. J. A., OEIS sequence A033879 – Deficiency of
*n*, or 2*n*−*σ*(*n*), https://oeis.org/A033879. - Sloane, N. J. A., & Guy, R. K., OEIS sequence A001065 – Sum of proper divisors (or aliquot parts) of
*n*: sum of divisors of*n*that are less than*n*, https://oeis.org/A001065. - Starni, P. (1991). On the Euler’s factor of an odd perfect number, Journal of Number Theory, 37, 366–369.
- Wikipedia contributors. (2019, March 6). Perfect number. In Wikipedia, The Free Encyclopedia. Retrieved from https://en.wikipedia.org/w/index.php?title=Perfect_number&oldid=886493275.

## Related papers

- Dris, J. A. B., & San Diego, I. T. (2020). Some modular considerations regarding odd perfect numbers – Part II.
*Notes on Number Theory and Discrete Mathematics*, 26 (3), 8-24, doi: 10.7546/nntdm.2020.26.3.8-24.

## Cite this paper

Dris, J. A. B., & San Diego, I. T. (2020). Some modular considerations regarding odd perfect numbers. *Notes on Number Theory and Discrete Mathematics*, 26 (2), 27-33, DOI: 10.7546/nntdm.2020.26.2.27-33.