On the Sum of Divisors function of linearly recurrent sequences

Let k 2 be an integer and k (n) be the sum of the kth powers of the positive divisors of n.In this paper, we prove that if (Un) n1 is any nondegenerate linearly recurrent sequence of integers whose general term is up to sign not a polynomial in n, then the inequality k (|Un|) |U k (n) | holds for all n sufficiently large, and the inequality (|Ufor almost all natural numbers n, where (n) = 1(n).In fact, the second inequality fails on a set of n x of counting function O(x/ log x).