On power values of sum of divisors function in arithmetic progressions

Sai Teja Somu () and Vidyanshu Mishra ()
Additional contact information
Sai Teja Somu: JustAnswer
Vidyanshu Mishra: Delhi Technological University

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 1, 335-340

Abstract: Abstract Let $$a\ge 1, b\ge 0$$ a ≥ 1 , b ≥ 0 and $$k\ge 2$$ k ≥ 2 be any given integers. It has been proven that there exist infinitely many natural numbers m such that sum of divisors of m is a perfect kth power. We try to generalize this result when the values of m belong to any given infinite arithmetic progression $$an+b$$ a n + b . We prove if a is relatively prime to b and order of b modulo a is relatively prime to k then there exist infinitely many natural numbers n such that sum of divisors of $$an+b$$ a n + b is a perfect kth power. We also prove that, in general, either sum of divisors of $$an+b$$ a n + b is not a perfect kth power for any natural number n or sum of divisors of $$an+b$$ a n + b is a perfect kth power for infinitely many natural numbers n.

Keywords: Sum of divisors; Power values; Arithmetic progressions; Primary: 11A25; Secondary: 11B25 (search for similar items in EconPapers)
Date: 2024
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s13226-023-00367-5 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:indpam:v:55:y:2024:i:1:d:10.1007_s13226-023-00367-5

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/13226

DOI: 10.1007/s13226-023-00367-5

Access Statistics for this article

Indian Journal of Pure and Applied Mathematics is currently edited by Nidhi Chandhoke

More articles in Indian Journal of Pure and Applied Mathematics from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().