On the least common multiple of Lucas subsequences (Q2867690)

Let \(A, B\) be fixed non-zero integers and denote by \((u_n)_{n\geq 0}\) a non-degenerate binary linear sequence with \(u_0=0\), \(u_1\neq 0\) and \(u_{n+2}=Au_{n+1}+Bu_n\) for all \(n\geq 0\). The authors of the present interesting paper evaluate (for \(n\to \infty\)) the quantity NEWLINE\[NEWLINE\frac{\log\left|\prod_{1\leq k\leq n,\; a_k\neq 0}\; u_{a_k}\right|}{\log \text{lcm}[u_{a_1},\ldots,u_{a_n}]},NEWLINE\]NEWLINE where \((a_n)_{n\geq 0}\) is either a fixed polynomial subsequence of the integers (namely, \(a_n=|f(n)|\) with \(f(X)\in\mathbb{Z}[X]\) under some suitable conditions), or some arithmetic function of \(n\) (such as the Euler totient function \(\varphi(n)\) or the sum of divisors function \(\sigma(n)\)), or where \((a_n)_{n\geq 0}\) is itself a binary recurrent sequence.NEWLINENEWLINEThe original result is due to \textit{Yu. V. Matiyasevich} and \textit{R. K. Guy} [Am. Math. Mon. 93, 631--635 (1986; Zbl 0614.10003)], who determined the limit in the case of Fibonacci numbers along the sequence of natural integers.