Arithmetic Properties of Certain t-Regular Partitions.

For a positive integer t ≥ 2 , let b t (n) denote the number of t-regular partitions of a nonnegative integer n. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo 2 for b 9 (n) and b 19 (n) . We prove some specific cases of two conjectures of Keith and Zanello on self-similarities of b 9 (n) and b 19 (n) modulo 2. For t ∈ { 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 b t (A n + B) ≡ 0 (mod 2) for all n ≥ 0 . We prove that, for any t ≥ 2 and prime ℓ , there are infinitely many arithmetic progressions A n + B for which ∑ n = 0 ∞ b t (A n + B) q n ≢ 0 (mod ℓ) . Next, we obtain quantitative estimates for the distributions of b 6 (n) , b 10 (n) and b 14 (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. [ABSTRACT FROM AUTHOR]