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The minimal excludant of an integer partition is the least positive integermissing from the partition. Let $\sigma_o\text{mex}(n)$ (resp.,$\sigma_e\text{mex}(n)$) denote the sum of odd (resp., even) minimal excludantsover all the partitions of $n$. Recently, Baruah et al. proved a fewcongruences for these partition functions modulo $4$ and $8$, and asked forasymptotic formulae for the same. In this article, we study the lacunarity of$\sigma_o\text{mex}(n)$ and $\sigma_e\text{mex}(n)$ modulo arbitrary powers of$2$ and also prove some infinite families of congruences for$\sigma_o\text{mex}(n)$ and $\sigma_e\text{mex}(n)$ modulo $4$ and $8$. We alsoobtain Hardy-Ramanujan type asymptotic formulae for both$\sigma_o\text{mex}(n)$ and $\sigma_e\text{mex}(n)$.

Arithmetic properties and asymptotic formulae for $\sigma_o\text{mex}(n)$ and $\sigma_e\text{mex}(n)$