Divisibility and distribution of $MEX$ related integer partitions of Andrews and Newman

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