Solutions of the problem of Erdös-Sierpiński: $σ(n)=σ(n+1)$
Abstract
Description
For $n\leq 1.5 \cdot 10^{10}$, we have found a total number of 1268 solutions to the Erdös-Sierpiński problem finding positive integer solutions of $σ(n)=σ(n+1)$, where $σ(n)$ 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 $σ(n)$, $n$ 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 $σ(n)=σ(n+1)=σ(n+k)=σ(n+k+1)$ 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 $n^{1/3}$. Finally some theorems related to the Erdös-Sierpiński problem are enunciated and proved.