Practical numbers in Lucas sequences

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 (un)n≥0 be the Lucas sequence satisfying u0 = 0, u1 = 1, and un+2 = aun+1 + bun for all integers n ≥ 0, where a and b are fixed nonzero integers. Assume a(b + 1) even and a 2 + 4b > 0. Also, let A be the set of all positive integers n such that |un| is a practical number. Melfi proved that A is infinite. We improve this result by showing that #A(x) ≫ x/ log x for all x ≥ 2, where the implied constant depends on a and b. We also pose some open questions regarding A.

Key words: Fibonacci numbers, Lucas sequences, practical numbers.