**Jose Arnaldo Bebita Dris**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 26, 2020, Number 3, Pages 25–32

DOI: 10.7546/nntdm.2020.26.3.25-32

**Full paper (PDF, 194 Kb)**

## Details

### Authors and affiliations

Jose Arnaldo Bebita Dris

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

### Abstract

In this note, we pursue an approach started in the M. Sc. thesis of the author and thereby attempt to produce stronger bounds for the sum *I*(*q ^{k}*) +

*I*(

*n*

^{2}), where

*q*

^{k}n^{2}is an odd perfect number with special prime

*q*and

*I*(

*x*) is the abundancy index of the positive integer

*x*.

### Keywords

- Odd perfect numbers
- Descartes–Frenicle–Sorli Conjecture
- Abundancy index

### 2010 Mathematics Subject Classification

- 11A05
- 11A25

### References

- Cohen, G. L., & Sorli, R. M. (2012). On Odd Perfect Numbers and Even 3-Perfect Numbers,
*Integers*, 12A, Article A6. - Dris, J. A. B., & Tejada, D.-J. U. (2018). Conditions equivalent to the Descartes–Frenicle–Sorli Conjecture on odd perfect numbers – Part II,
*Notes Number Theory Discrete Math*., 24 (3), 62–67. - Dris, J. A. B. (2017). The abundancy index of divisors of odd perfect numbers Part III,
*Notes Number Theory Discrete Math*., 23 (3), 53–59. - Dris, J. A. B. (2017). Conditions equivalent to the Descartes–Frenicle–Sorli Conjecture on odd perfect numbers,
*Notes Number Theory Discrete Math*., 23 (2), 12–20. - Dris, J. A. B. (2017). Analysis of the ratio
*D*(*n*)/*n*, preprint, https://arxiv.org/abs/1703.09077. - Dris, J. A. B. (2012). The abundancy index of divisors of odd perfect numbers,
*J. Integ. Seq.*, 15 (4), Article 12.4.4. - Dris, J. A. B. (2008).
*Solving the odd perfect number problem: some old and new approaches*, M. Sc. thesis, De La Salle University, Manila, Philippines. - Malyshev, A. V. (2001). [1994], “Quadratic form”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php/Quadratic_form.
- MSE user mathlove and Dris, J. A. B. (2018). Global extrema for (qk−1)(qk+1−2qk+1) qk(q−1)(qk+1−1) ?, https://math.stackexchange.com/questions/2998091, Last updated on 11/15/2018.

## Related papers

- Dris, J. A. B., & Tejada, D.-J. U. (2018). Conditions equivalent to the Descartes–Frenicle–Sorli Conjecture on odd perfect numbers – Part II,
*Notes Number Theory Discrete Math*., 24 (3), 62–67. - Dris, J. A. B. (2017). The abundancy index of divisors of odd perfect numbers Part III,
*Notes Number Theory Discrete Math*., 23 (3), 53–59. - Dris, J. A. B. (2017). Conditions equivalent to the Descartes–Frenicle–Sorli Conjecture on odd perfect numbers,
*Notes Number Theory Discrete Math*., 23 (2), 12–20.

## Cite this paper

Dris, J. A. B. (2020). On the quantity *I*(*q ^{k}*) +

*I*(

*n*

^{2}) where

*q*is an odd perfect number.

^{k}n^{2}*Notes on Number Theory and Discrete Mathematics*, 26(3), 25-32, DOI: 10.7546/nntdm.2020.26.3.25-32.