The minimal excludant or mex of a partition, as introduced by Andrews and Newman [3], is the smallest positive integer missing from that partition. Recently, Ballantine and Merca [5] studied a generalization of mex called the least r-gap and defined as the smallest positive integer which does not appear at least r times in the partition. In this article, we deduce the generating functions for certain arithmetic functions related to the least r-gaps and establish their connections with known partition functions. We also obtain arithmetic properties and asymptotic formulae for some of these functions. As a consequence of one such result, we find an asymptotic formula for the Andrews’ singular overpartition function \({\overline{C}}_{k,i}(n)\) .
