Erdős Problem #710

- $78

Let $f(n)$ be minimal such that in $(n,n+f(n))$ there exist distinct integers $a_1,\ldots,a_n$ such that $k\mid a_k$ for all $1\leq k\leq n$. Obtain an asymptotic formula for $f(n)$.

#710: [ErPo80][Er92c]

number theory

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A problem of Erdős and Pomerance [ErPo80], who proved\[(2/\sqrt{e}+o(1))n\left(\frac{\log n}{\log\log n}\right)^{1/2}\leq f(n)\leq (1.7398\cdots+o(1))n(\log n)^{1/2}.\]In [Er92c] Erdős offered 2000 rupees for an asymptotic formula; for uniform comparison across prizes I have converted this using the 1992 exchange rates.

See also [711].

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Related OEIS sequences: A390246

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Additional thanks to: Wouter van Doorn

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T. F. Bloom, Erdős Problem #710, https://www.erdosproblems.com/710, accessed 2026-02-13