Andrews and Newman introduced the mex-function mexA,a(λ)\documentclass[12pt]{minimal}
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\begin{document}$$\text {mex}_{A,a}(\lambda )$$\end{document} for an integer partition λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document} of a positive integer n as the smallest positive integer congruent to a modulo A that is not a part of λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document}. They then defined pA,a(n)\documentclass[12pt]{minimal}
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\begin{document}$$p_{A,a}(n)$$\end{document} to be the number of partitions λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document} of n satisfying mexA,a(λ)≡a(mod2A)\documentclass[12pt]{minimal}
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\begin{document}$$\text {mex}_{A,a}(\lambda )\equiv a\pmod {2A}$$\end{document}. They found the generating function for pt,t(n)\documentclass[12pt]{minimal}
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\begin{document}$$p_{t,t}(n)$$\end{document} and p2t,t(n)\documentclass[12pt]{minimal}
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\begin{document}$$p_{2t,t}(n)$$\end{document} for any positive integer t, and studied their arithmetic properties for some small values of t. In this article, we study the partition function pmt,t(n)\documentclass[12pt]{minimal}
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\begin{document}$$p_{mt,t}(n)$$\end{document} for all positive integers m and t. We show that for sufficiently large X, the number of all positive integers n≤X\documentclass[12pt]{minimal}
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\begin{document}$$n\le X$$\end{document} such that pmt,t(n)\documentclass[12pt]{minimal}
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\begin{document}$$p_{mt,t}(n)$$\end{document} is an even number is at least O(X/3)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {O}}(\sqrt{X/3})$$\end{document} for all positive integers m and t. We also prove that for sufficiently large X, the number of all positive integers n≤X\documentclass[12pt]{minimal}
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\begin{document}$$n\le X$$\end{document} such that pmp,p(n)\documentclass[12pt]{minimal}
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\begin{document}$$p_{mp,p}(n)$$\end{document} is an odd number is at least O(loglogX)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {O}}(\log \log X)$$\end{document} for all m≢0(mod3)\documentclass[12pt]{minimal}
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\begin{document}$$m\not \equiv 0\pmod {3}$$\end{document} and all primes p≡1(mod3)\documentclass[12pt]{minimal}
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\begin{document}$$p\equiv 1\pmod {3}$$\end{document}. Finally, we establish identities connecting the ordinary partition function to pmt,t(n)\documentclass[12pt]{minimal}
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\begin{document}$$p_{mt,t}(n)$$\end{document}.
