A note on the restricted partition function pA(n, k)

Let A = (a(n))(n is an element of)N+ be a sequence of positive integers. Let p(A)(n, k) denote the number of multi-color partitions of n into parts in {a(1), ... , a(k)}. We examine several arithmetic properties of the sequence (p(A)(n, k) (mod m))n is an element of N for an arbitrary fixed integer m >= 2. We investigate periodicity of the sequence and lower and upper bounds for the density of the set {n is an element of N : p(A)(n, k)equivalent to i (mod m)} for a fixed positive integer k and i is an element of {0, 1, ..., m(-1)}. In particular, we apply our results to the special cases of the sequence A. Furthermore, we present some results related to restricted m-ary partitions. (C) 2022 Elsevier B.V. All rights reserved.