No abstract

Let m ∈ N ≥2 and for given k ∈ N + consider the sequence (A m,k (n)) n∈N defined by the power series expansionThe number A m,k (n) counts the number of representations of n as sums of powers of m, where each summand has one among k colors. In this note we prove that for each p ∈ P ≥3 and s ∈ N + , the p-adic valuation of the number A p,( p−1)( p s −1) (n) is equal to 1 for n ≥ p s . We also obtain some results concerning the behaviour of the sequence (ν p (A p,( p−1)(up s −1) (n))) n∈N for fixed u ∈ {2, . . . , p − 1} and p ≥ 3. Our results generalize the earlier findings obtained for p = 2 by Gawron, Miska and the first author.

Let k ∈ N ≥2 and for given m ∈ Z\{0} consider the sequence (S k,m (n)) n∈N defined by the power series expansion 1 (1 − x) m ∞ i=0 1 (1 − x k i) m(k−1) = ∞ n=0 S k,m (n)x n. The number S k,m (n) for m ∈ N + has a natural combinatorial interpretation: it counts the number of representations of n as sums of powers of k, where the part equal to 1 takes one among mk colors and each part > 1 takes m(k − 1) colors. We concentrate on the case when k = p is a prime. Our main result is the computation of the exact value of the p-adic valuation of S p,m (n). In particular, in each case the set of values of ν p (S p,m (n)) is finite and the maximum value is bounded by max{ν p (m) + 1, ν p (m + 1) + 1}. Our results can be seen as a generalization of earlier work of Churchhouse and recent work of Gawron, Miska and Ulas, and the present authors.