A process for producing a polymer, which process comprises reacting at least ethylene, 1 -pentene and 1 -hexene in at least one reaction zone, while maintaining the reaction zone at a pressure between atmospheric pressure and 60 kg/cm2 and at a temperature between 30° C. and 120° C., in the presence of a catalyst system, to produce a polymerized product having a polyethyelene chain, with the catalyst system comprising a titanium based Ziegler-Natta catalyst, and, as a co-catalyst, an aluminium compound, with the catalyst having the ability to incorporate simultaneously about the same level of 1-pentene and 1-hexene in the polyethylene chain.