Glasnik Matematicki, Vol. 56, No. 1 (2021), 17-28.

SOME ARITHMETIC FUNCTIONS OF FACTORIALS IN LUCAS SEQUENCES

Eric F. Bravo and Jhon J. Bravo

Departamento de Matemáticas, Universidad del Cauca, Calle 5 No. 4-70 Popayán, Colombia
e-mail: fbravo@unicauca.edu.co
e-mail: jbravo@unicauca.edu.co

Abstract.   We prove that if {un}n≥ 0 is a nondegenerate Lucas sequence, then there are only finitely many effectively computable positive integers n such that |un|=f(m!), where f is either the sum-of-divisors function, or the sum-of-proper-divisors function, or the Euler phi function. We also give a theorem that holds for a more general class of integer sequences and illustrate our results through a few specific examples. This paper is motivated by a previous work of Iannucci and Luca who addressed the above problem with Catalan numbers and the sum-of-proper-divisors function.

2010 Mathematics Subject Classification.   11A25, 11B39

Key words and phrases.   Lucas sequence, arithmetic function, Diophantine equation

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https://doi.org/10.3336/gm.56.1.02

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