In this work we consider the cognitive many-to-one interference network. We first extend existing coding schemes from the two-user case to this network scenario. Then we present a novel coding scheme using compute-and-forward and show it can enlarge the achievable rate region considerably for a wide range of parameters. Numerical evaluations are given to compare the performance of different schemes. Specializing the results to symmetric settings, for a range of parameters, our achievable rate region is shown to be within a constant gap from capacity, regardless of the number of cognitive users.