Infinite families of infinite families of congruences for k-regular partitions

Let k∈{10,15,20}, and let b k (n) denote the number k-regular partitions of n. We prove for half of all primes p and any t≥1 that there exist p−1 arithmetic progressions modulo p 2t such that b k (n) is a multiple of 5 for each n in one of these progressions.