S-parts of sums of terms of linear recurrence sequences

Abstract

Let $S= \{ p_1, \ldots, p_s\}$ be a finite, non-empty set of distinct prime numbers and $(U_{n})_{n \geq 0}$ be a linear recurrence sequence of integers of order at least 2. For any positive integer k, and $w = (w_k, \ldots, w_1)\in\mathbb{Z}^k, w_1, \ldots, w_k\neq 0$ we define $(U_j^{(k, w)})_{j\geq 1}$ an increasing sequence composed of integers of the form $|w_kU_{n_k} +\cdots + w_1U_{n_1}|$, $n_k>\cdots >n_1$. Under certain assumptions, we prove that for any $\varepsilon >0$, there exists an integer $n_{0}$ such that $[U_j^{(k,w)}]_S < (U_j^{(k, w)})^{\varepsilon},$ ${\rm for}\, j > n_0$, where $[m]_S$ denotes the S-part of the positive integer m. On further assumptions on $(U_{n})_{n \geq 0}$, we also compute an effective bound for $[U_j^{(k, w)}]_S$ of the form $(U_j^{(k,w)})^{1-c}$, where c is a positive constant depending only on r, $a_1$ , . . ., $a_r$, $U_0$, . . ., $U_{r-1}$ , $w_1$, . . ., $w_k$ and S.