number-theory / Number theory
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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Append-only history
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.
Research memory
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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