Let \(b_t(n)\) denote the number of t-regular partitions of n. In recent years, a number of congruences modulo powers of 2 for \(b_t(n)\) have been proved for some small t. Ballantine and Merca proved some congruences modulo powers of 2 for \(b_4(n)\) . Cherubini and Mercuri presented a characterization of the parity of \(b_8(n)\) . Cui and Gu established infinite families of congruences modulo 2 for \(b_{16}(n)\) . Motivated by their works, we present two characterizations of congruences modulo 4 for \(b_4(n)\) and the parity of \(b_{16}(n)\) . In particular, we confirm three conjectures on self-similar congruence properties of \(b_4(n)\) , \(b_8(n)\) and \(b_{16}(n)\) posed by Du and Tang.
