On Bounds of Non-Deficient Numbers

Uma Dixit

doi:10.17485/IJST/v15i35.946

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Indian Journal of Science and Technology

DOI: 10.17485/IJST/v15i35.946

Year: 2022, Volume: 15, Issue: 35, Pages: 1683-1690

Original Article

On Bounds of Non-Deficient Numbers

Uma Dixit^1*

¹Department of mathematics, University postgraduate college secunderabad, Osmania university, Hyderabad, 500003, Telangana, India

*Corresponding Author
Email: [email protected]

Received Date:05 March 2022, Accepted Date:11 July 2022, Published Date:03 September 2022

This work is licensed under a Creative Commons Attribution 4.0 International License.

Abstract

Objectives: To improve the upper bounds of a quasi perfect number and give an important result on its divisibility with primes. Methods: A positive integer n is quasi perfect if s (n) >2n + 1, where s (n) denotes the sum of the positive divisors of n. However, the existence of a quasi perfect number, which is a Non-Deficient number, is still an open problem. We use R(n), the sum of the reciprocals of distinct primes dividing the quasi perfect number, to derive lemmas and improve the bounds obtained by earlier authors. Findings: We improve the upper bounds for R(n), when n is quasi perfect with gcd (15, n) = 3 or gcd (15, n) = 5. As a consequence, we establish that a quasi perfect number, if exists, is divisible by both 3 and 5 or by none of them. Novelty: The unique method of using R(n) also resulted in finding an important result that 3, 5 and 7 cannot divide any quasi perfect number. Mathematics Subject Classification: 11A05, 11A25.

Keywords: non-deficient number; quasi perfect number; sum of the divisor; sum of the reciprocal; bounds of perfect number; number of divisors

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Copyright

© 2022 Dixit. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Published By Indian Society for Education and Environment (iSee)