The additive number theory is a subdiscipline of number theory dealing with problems of sums in the set $\mathbb Z$ of integers, sometimes also only positive integers, i.e. the set $\mathbb N$ of natural numbers. There are many famous problems from this subdiscipline, some of which are still unsolved, including the Goldbach conjecture. We start with some elementary results concerning special sums of integers, namely the sum of divisors of a given natural number $n > 0.$