RAMANUJAN-TYPE CONGRUENCES MODULO POWERS OF 5 AND 7

Let b(l)(n) denote the number of l-regular partitions of n. In 2012, using the theory of modular forms, Furcy and Penniston presented several infinite families of congruences modulo 3 for some values of l. In particular, they showed that for alpha, n >= 0, b(25) (3(2 alpha+3) n+2 . 3(2 alpha+2)-1) equivalent to 0 (mod 3). Most recently, congruences modulo powers of 5 for c(5)(n) was proved by Wang, where c(N)(n) counts the number of bipartitions (lambda(1),lambda(2)) of n such that each part of lambda(2) is divisible by N. In this paper, we prove some interesting Ramanujan-type congruences modulo powers of 5 for b(25)(n), B-25(n), c(25)(n) and modulo powers of 7 for c(49)(n). For example, we prove that for j >= 1, c(25) (5(2j)n + 11.5(2j) + 13/12) equivalent to 0 (mod 5(j+1)), c(49) (7(2j)n + 11.7(2j) + 25/12) equivalent to 0 (mod 7(j+1)) and b(25) (3(2 alpha+3) . 5(2j-1). n + 2 . 3(2 alpha+2) . 5(2j-1) - 1) equivalent to 0 (mod 3 . 5(2j-1)).