On the quantity I(qk) + I(n2) where qk n2 is an odd perfect number

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
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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(qk) + I(n2), where qkn2 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

  1. Cohen, G. L., & Sorli, R. M. (2012). On Odd Perfect Numbers and Even 3-Perfect Numbers, Integers, 12A, Article A6.
  2. 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.
  3. 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.
  4. 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.
  5. Dris, J. A. B. (2017). Analysis of the ratio D(n)/n, preprint, https://arxiv.org/abs/1703.09077.
  6. Dris, J. A. B. (2012). The abundancy index of divisors of odd perfect numbers, J. Integ. Seq., 15 (4), Article 12.4.4.
  7. 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.
  8. 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.
  9. 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.

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Cite this paper

Dris, J. A. B. (2020). On the quantity I(qk) + I(n2) where qk n2 is an odd perfect number. Notes on Number Theory and Discrete Mathematics, 26 (3), 25-32, doi: 10.7546/nntdm.2020.26.3.25-32.

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