For example,\[\sum_{3\leq n\leq 5}\frac{1}{n} = \frac{47}{60}\textrm{ and }\sum_{3\leq n\leq 6}\frac{1}{n}=\frac{19}{20}.\]The smallest $b$ for each $a$ are listed at A375081 at the OEIS.

This was resolved in the affirmative by van Doorn [vD24], who proved $b=b(a)$ always exists, and in fact $b(a) \ll a$. Indeed, if $a\in (3^k,3^{k+1}]$ then one can take $b=2\cdot 3^{k+1}-1$. van Doorn also proves that $b(a)>a+(1/2-o(1))\log a$, and considers various generalisations of the original problem.

van Doorn proves, more precisely, that $b(a)<4.374a$ for all $a>1$, and that $b(a)>a+0.54\log a$ for all large $a$. He further proves that $b(a)<a+0.61\log a$ for infinitely many $a$, and expects that there are infinitely many $a$ with $b(a)>a+(\log a)^2$. It seems likely that $b(a)\leq (1+o(1))a$, and perhaps even $b(a)\leq a+(\log a)^{O(1)}$.

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This page was last edited 28 December 2025.