We investigate the question of optimal input ensembles for memory channels and construct a rather large class of Pauli channels with correlated noise, which can be studied analytically with regard to the entanglement of their optimal input ensembles. In a more detailed study of a subclass of these channels, we obtain the complete phase diagram of the two-qubit channel . A more concrete model, based on random rotations of the error operators which mimic the behavior of this subclass of channels, will be a presented and we see that the natural extension of this memory channel to more than two uses is possible . As an instance of this extension we study Bosonic memory channel with additive noise and address the weak forgetfulness of this channel. We find the classical capacity of this channel by unraveling the correlation and using the know results about memoryless channels.