Divisibility properties by recurrence relations

We present applications to a fairly general criterion to obtain divisibility properties of a sequence defined by a linear recurrence with coefficients satisfying some divisibility patterns. Let nu(p)(m) denote the exponent of the highest power of a prime p which divides m. This number is often referred to as the p-adic order of m. We determine nu(p) (Sigma(k=0)(infinity) ((k p))a(k)) in terms of n for an integer a by the method which offers insight into the structure of the problem without explicitly calculating the coefficients of the related recurrence. We find that the p-adic order of these sums depends on nu(p)(a + 1) for a prime p greater than or equal to 3 and on nu(2)(a - 1) for p = 2.