Count data is often found in many real applications and some observations may occur in the data set in an excessive amount. In many real problems it is quite common for the data set to contain excesses of zero and one observations. In a more general context, k1 and k2 are defined as observations of a particular data set that have discrepancy (excess) in their frequencies, making modeling from traditional discrete distributions inappropriate. Thus, the main objective of this work is to propose the k1 and k2 Inflated Power Series family of distributions, aiming to model data sets that present such discrepancy in the observations k1 and k2. In order to estimate the parameters we consider classical approach, with the maximum likelihood method, using a hurdle version of distributions. Some applications considering real data sets will be presented.