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On power values of sum of divisors function in arithmetic progressions

Sai Teja Somu () and Vidyanshu Mishra ()
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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
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DOI: 10.1007/s13226-023-00367-5

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