Solutions of the problem of Erd\"os-Sierpi\'nski: $\sigma (n)=\sigma (n+1)$

For $n\le 1.5\cdot {10}^{10}$, we have found a total number of 1268 solutions to the Erd"os-Sierpi'nski problem finding positive integer solutions of $\sigma (n)=\sigma (n+1)$, where $\sigma (n)$ is the sum of the positive divisors of n. On the basis of that set of solutions the following empirical properties are enunciated: first, all the $\sigma (n)$, $n$ being a solution, are divisible by 6; second, the repetition of solutions leads to the formulation of a new problem: \emph{Find the natural numbers $n$ such that $\sigma (n)=\sigma (n+1)=\sigma (n+k)=\sigma (n+k+1)$ for some positive integer $k$}. A third empirical property concerns the asymptotic behavior of the function of $n$ that gives the number of solutions for $m$ less or equal to $n$, which we find to be as $n^{1\mathrm{/}3}$. Finally some theorems related to the Erd"os-Sierpi'nski problem are enunciated and proved.