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 a2 + 4b > 0. Also, let (Figure presented.) be the set of all positive integers n such that |un| is a practical number. Melfi proved that (Figure presented.) is infinite. We improve this result by showing that # (Figure presented.) (x) ≫ x/log x for all x ≥ 2, where the implied constant depends on a and b. We also pose some open questions regarding (Figure presented.).

Practical numbers in Lucas sequences / Sanna, C.. - In: QUAESTIONES MATHEMATICAE. - ISSN 1607-3606. - STAMPA. - 42:7(2019), pp. 977-983. [10.2989/16073606.2018.1502697]

### 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 a2 + 4b > 0. Also, let (Figure presented.) be the set of all positive integers n such that |un| is a practical number. Melfi proved that (Figure presented.) is infinite. We improve this result by showing that # (Figure presented.) (x) ≫ x/log x for all x ≥ 2, where the implied constant depends on a and b. We also pose some open questions regarding (Figure presented.).
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11583/2789392`