Arithmetic Properties of Certain t-Regular Partitions

For a positive integer t > 2, let bt(n) denote the number of t-regular partitions of a nonnegative integer n. Motivated by some re-cent conjectures of Keith and Zanello, we establish infinite families of congruences modulo 2 for b9(n) and b19(n). We prove some specific cases of two conjectures of Keith and Zanello on self-similarities of b9(n) and b19(n) modulo 2. For t E {6,10, 14, 15, 18, 20, 22, 26, 27, 28}, Keith and Zanello conjectured that there are no integers A > 0 and B > 0 for which bt(An + B) 0 (mod 2) for all n > 0. We prove that, for any t > 2 and prime $, there are infinitely many arithmetic progressions An + B for which S8n=0 bt(An +B)qn ?0 (mod $). Next, we obtain quantitative estimates for the distributions of b6(n), b10(n) and b14(n) modulo 2. We further study the odd densities of certain infinite families of eta-quotients related to the 7-regular and 13-regular partition functions.