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

Abstract

Let $(L_n{)}_{n\ge 1}$ 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 $n\ge 1$. We determine the asymptotic behavior of $\log \mathrm{lcm}(L_1+s_1,L_2+s_2,\dots ,L_n+s_n)$ as $n\to +\infty$, for $(s_n{)}_{n\ge 1}$ a periodic sequence in $\{-1,+1\}$. We also carry out the same analysis for $(s_n{)}_{n\ge 1}$ 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.