Ramanujan-type congruences modulo powers of 5 and 7

Let b _? (n) denote the number of ?-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 ?. In particular, they showed that for α, n ≥ 0, b ₂₅ (3^2α+3 n+2 · 3^2α+2-1) ≡ 0 (mod 3). Most recently, congruences modulo powers of 5 for c5(n) was proved by Wang, where c _N (n) counts the number of bipartitions (λ₁,λ₂) of n such that each part of λ₂ is divisible by N. In this paper, we prove some interesting Ramanujan-type congruences modulo powers of 5 for b25(n), B25(n), c25(n) and modulo powers of 7 for c49(n). For example, we prove that for j ≥ 1, ({c_{25}}left( {{5^{2j}}n + frac{{11 cdot {5^{2j}} + 13}}{{12}}} right) equiv 0) (mod 5 ^j+1), ({c_{49}}left( {{7^{2j}}n + frac{{11 cdot {7^{_{2j}}} + 25}}{{12}}} right) equiv 0) (mod 7 ^j+1) and b ₂₅ (3^2α+3 · n+2 · 3^2α+2-1) ≡ 0 (mod 3 · 5^2j-1).