Erdös-Sierpienski问题的解：$σ(n)=σ(n+1)$
Solutions of the problem of Erdös-Sierpiński: $σ(n)=σ(n+1)$
摘要：
对于$n\leq 1.5 \cdot 10^{10}$，我们已经找到了总数为1268的解决方案的Erdös-Sierpienski问题寻找正整数解的$σ(n)=σ(n+1)$，其中$σ(n)$是正因子的总和n。在这组解的基础上，阐明了以下经验性质：第一，所有的$σ(n)$，$n$是一个解，都能被6整除;第二，解的重复导致一个新问题的形成：\emph{Find the natural numbers $n$ such that $σ(n)=σ(n+1)=σ(n+k)=σ(n+k+1)$ for some positive integer $k$}。第三个经验性质涉及$n$函数的渐近行为，该函数给出了$m$的解的个数小于或等于$n$，我们发现它是$n^{1/3}$。最后给出并证明了与Erdös-Sierpienski问题有关的几个定理。
英文摘要：
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.
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