Andrews and Newman introduced the minimal excludant or ``$mex$'' function for an integer partition $π$ of a positive integer $n$, $mex(π)$, as the smallest positive integer that is not a part of $π$. They defined $σmex(n)$ to be the sum of $mex(π)$ taken over all partitions $π$ of $n$. We prove infinite families of congruence and multiplicative formulas for $σmex(n)$. By restricting to the part of $π$, Andrews and Newman also introduced $moex(π)$ to be the smallest odd integer that is not a part of $π$ and $σmoex(n)$ to be the sum of $moex(π)$ taken over all partitions $π$ of $n$. In this article, we show that for any sufficiently large $X$, the number of all positive integer $n\leq X$ such that $σmoex(n)$ is an even (or odd) number is at least $\mathcal{O}(\log \log X)$.