New congruences modulo powers of 2 for broken 3-diamond partitions and 7-core partitions

Let Δk(n)Δk(n) denote the number of broken k-diamond partitions of n for a fixed positive integer k  . Recently, Radu and Sellers conjectured that for all α⩾1α⩾1 and n⩾0n⩾0, Δ3(λα)Δ3(2α+2n+λα+2)≡Δ3(λα+2)Δ3(2αn+λα)(mod2α), where λα=2α+1+13 if α   is even and λα=2α+13 if α   is odd. Radu and Sellers proved that this conjecture is true for α=1α=1. In this work, we show that this conjecture holds for α=2α=2. We also prove that Δ3(λα)≡(−1)[α2](mod4) which yields Δ3(λα)≡1(mod2). This congruence was conjectured by Radu and Sellers. Furthermore, we also deduce some new Ramanujan-type congruences modulo 2 and 4 for 7-core partitions.