This paper concerns power series of an arithmetic nature that arise in the analysis of divide-and-conquer algorithms. Two key notions are studied : that of B-regular sequence and that of Malherian sequence with their associated power series. Firstly we emphasize the link between rational series over the alphabet {x0, x1,....,xB-1} and B-regular series. Secondly we extend the theorem of Christol, Kamae, Mendes France and Rauzy about automatic sequences and algebraic series to B-regular sequences and Malherian series. We develop here a construtive theory of B-regular and Malherian series. The examples show the ubiquitous character of B-regular series in the study of arithmetic functions related to number representation systems and divide-and-conquer algorithms.