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]
Titolo: | Practical numbers in Lucas sequences | |
Autori: | ||
Data di pubblicazione: | 2019 | |
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Digital Object Identifier (DOI): | http://dx.doi.org/10.2989/16073606.2018.1502697 | |
Appare nelle tipologie: | 1.1 Articolo in rivista |
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http://hdl.handle.net/11583/2789392