A minimal r-complete partition of an integer m is a partition of m with as few parts as possible, such that all the numbers 1,..., rm can be written as a sum of parts taken from the partition, each part being used at most r times. This is a generalization of M-partitions (minimal 1-complete partitions). The number of M-partitions of m was recently connected to the binary partition function and two related arithmetic functions. In this paper we study the case r ≥ 2, and connect the number of minimal r-complete partitions to the (r+1)-ary partition function and a related arithmetic function.