Here we assume a discrete random variable, possessing a one-to-one correspondence with the set of natural numbers. Its Shannon entropy is considered and some distributions, obtained by means of the Maximum Entropy Principle, will be discussed too. Then two entropies, the q-entropy and κ-entropy proposed by C. Tsallis and G. Kaniadakis respectively, will be considered. These entropies have as their limit the Shannon entropy when entropic parameters q and κ approach specific values. We will show some relationships existing between the Shannon entropy and these generalized entropies and between Tsallis and Kaniadakis entropies and give some links regarding other functions, in particular logarithms. Exponentials will be discussed too. We will also address ourselves to the generalization of the sum, in the framework of the κ-calculus proposed by Kaniadakis.