More Congruences for the k-regular Partition Function

Abstract

A partition of a number n is a list of positive integers whose sum is n. For example, 4 + 2 + 1 and 4 + 1 + 1 + 1 are both partitions of 7. It can be shown that 4 has 5 partitions, 9 has 30 partitions, 14 has 135 partitions, and Srinivasa Ramanujan proved the following beautiful result: the number of partitions of 5n + 4 is divisible by 5 for any nonnegative integer n. The k-regular partition function counts the number of partitions of n whose parts are not divisible by k. In 2012, for particular values of k, David Furcy and David Penniston found many families of integers whose number of k-regular partitions is divisible by 3. In this paper, I extend their results to larger values of k and provide an overview of the methodology used to arrive at the result. In the interest of brevity, only a sketch of the proof is given.