On the Sum of Divisors function of linearly recurrent sequences

Lucaa, Florian and Campbell Manape, Makoko (2025) On the Sum of Divisors function of linearly recurrent sequences. Annales Mathematicae et Informaticae, 62. pp. 91-98. ISSN 1787-5021 (Print) 1787-6117 (Online)

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Abstract

Let k ≥ 2 be an integer and σk(n) be the sum of the kth powers of the positive divisors of n. In this paper, we prove that if (Un)n≥1 is any nondegenerate linearly recurrent sequence of integers whose general term is up to sign not a polynomial in n, then the inequality σk(|Un|) ≤ |Uσk(n)| holds for all n sufficiently large, and the inequality σ(|Unk|) ≤ |Uσk(n)| holds for almost all natural numbers n, where σ(n) = σ1(n). In fact, the second inequality fails on a set of n ≤ x of counting function O(x/logx)

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