Erdős problem 1061 · problems.science

For $S(x) = \#\{(a,b) : a + b \le x,\ \sigma(a) + \sigma(b) = \sigma(a+b)\}$ , is $S(x) \sim cx$ ? The preprint claims $S(x)$ grows faster than $x (\log x)^R$ for every fixed $R$ , ruling out the linear asymptotic.

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Let $S(x)$ count ordered pairs $(a,b)$ with $a+b \le x$ and $\sigma(a)+\sigma(b) = \sigma(a+b)$. Erdos asked whether $S(x) \sim cx$. The opposite extreme holds: for every $R > 0$, $S(x)/(x(\log x)^R) \to \infty$, so the count beats every fixed logarithmic scale.

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