$S$-parts of sums of terms of linear recurrence sequences

Abstract

Let $S=\{p_1,\dots ,p_s\}$ be a finite, non-empty set of distinct prime numbers and $(U_n{)}_{n\ge 0}$ be a linear recurrence sequence of integers of order $r$. For any positive integer $k,$ we define $(U_j^{(k)}{)}_{j\ge 1}$ an increasing sequence composed of integers of the form $U_{n_k}+\cdots +U_{n_1},n_k>\cdots >n_1$. Under certain assumptions, we prove that for any $\epsilon >0,$ there exists an integer $n_0$ such that $[U_j^{(k)}{]}_S<{\left(U_j^{(k)}\right)}^{\epsilon },$ for $j>n_0,$ where $[m{]}_S$ denote the $S$-part of the positive integer $m$. On further assumptions on $(U_n{)}_{n\ge 0},$ we also compute an effective bound for $[U_j^{(k)}{]}_S$ of the form ${\left(U_j^{(k)}\right)}^{1-c}$, where $c$ is a positive constant depends only on $(U_n{)}_{n\ge 0}$ and $S.$