Article

Practical numbers in Lucas sequences

Published in: Quaestiones Mathematicae
Volume 42 , issue 7 , 2019 , pages: 977–983

DOI: 10.2989/16073606.2018.1502697

Author(s): Carlo Sanna Università degli Studi di Torino, Italy

Keywords: Primary: 11B37 , Secondary: 11B39 , 11N25 , Primary: 11B37 , Secondary: 11B39 , 11N25

Abstract

A practical number is a positive integer n such that all the positive integers m ≤ n can be written as a sum of distinct divisors of n. Let (u_n)_n_≥0 be the Lucas sequence satisfying u₀ = 0, u₁ = 1, and u_n₊₂ = au_n₊₁ + bu_n for all integers n ≥ 0, where a and b are fixed nonzero integers. Assume a(b + 1) even and a² + 4b > 0. Also, let be the set of all positive integers n such that |u_n| is a practical number. Melfi proved that is infinite. We improve this result by showing that #(x) ≫ x/log x for all x ≥ 2, where the implied constant depends on a and b. We also pose some open questions regarding .

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