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\begin{document}$$t\ge 2$$\end{document}, let bt(n)\documentclass[12pt]{minimal}
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\begin{document}$$b_{t}(n)$$\end{document} 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 b9(n)\documentclass[12pt]{minimal}
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\begin{document}$$b_9(n)$$\end{document} and b19(n)\documentclass[12pt]{minimal}
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\begin{document}$$b_{19}(n)$$\end{document}. We prove some specific cases of two conjectures of Keith and Zanello on self-similarities of b9(n)\documentclass[12pt]{minimal}
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\begin{document}$$b_9(n)$$\end{document} and b19(n)\documentclass[12pt]{minimal}
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\begin{document}$$b_{19}(n)$$\end{document} modulo 2. For t∈{6,10,14,15,18,20,22,26,27,28}\documentclass[12pt]{minimal}
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\begin{document}$$t\in \{6,10,14,15,18,20,22,26,27,28\}$$\end{document}, Keith and Zanello conjectured that there are no integers A>0\documentclass[12pt]{minimal}
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\begin{document}$$A>0$$\end{document} and B≥0\documentclass[12pt]{minimal}
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\begin{document}$$B\ge 0$$\end{document} for which bt(An+B)≡0(mod2)\documentclass[12pt]{minimal}
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\begin{document}$$b_t(An+ B)\equiv 0\pmod 2$$\end{document} for all n≥0\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 0$$\end{document}. We prove that, for any t≥2\documentclass[12pt]{minimal}
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\begin{document}$$t\ge 2$$\end{document} and prime ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document}, there are infinitely many arithmetic progressions An+B\documentclass[12pt]{minimal}
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\begin{document}$$An+B$$\end{document} for which ∑n=0∞bt(An+B)qn≢0(modℓ)\documentclass[12pt]{minimal}
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\begin{document}$$\sum _{n=0}^{\infty }b_t(An+B)q^n\not \equiv 0 \pmod {\ell }$$\end{document}. Next, we obtain quantitative estimates for the distributions of b6(n),b10(n)\documentclass[12pt]{minimal}
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\begin{document}$$b_{6}(n), b_{10}(n)$$\end{document} and b14(n)\documentclass[12pt]{minimal}
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\begin{document}$$b_{14}(n)$$\end{document} 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.
