Arithmetic properties and asymptotic formulae for σomex(n) and σemex(n)

The minimal excludant of an integer partition is the least positive integer missing from the partition. Let sigma(o)mex(n) (resp., sigma(e)mex(n)) denote the sum of odd (resp., even) minimal excludants over all the partitions of n. Recently, Baruah et al. proved a few congruences for these partition functions modulo 4 and 8, and asked for asymptotic formulae for the same. In this article, we study the lacunarity of sigma(o)mex(n) and sigma(e)mex(n) modulo arbitrary powers of 2 and also prove some infinite families of congruences for sigma(o)mex(n) and sigma(e)mex(n) modulo 4 and 8