On the l.c.m. of shifted Lucas numbers

Abstract

Let $(L_n)_{n \geq 1}$ be the sequence of Lucas numbers, defined recursively by $L_1 := 1$, $L_2 := 3$, and $L_{n + 2} := L_{n + 1} + L_n$, for every integer $n \geq 1$. We determine the asymptotic behavior of $\log \operatorname{lcm} (L_1 + s_1, L_2 + s_2, \dots, L_n + s_n)$ as $n \to +\infty$, for $(s_n)_{n \geq 1}$ a periodic sequence in $\{-1, +1\}$. We also carry out the same analysis for $(s_n)_{n \geq 1}$ a sequence of independent and uniformly distributed random variables in $\{-1, +1\}$. These results are Lucas numbers-analogs of previous results obtained by the author for the sequence of Fibonacci numbers.