Problem 269

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Statement. Let $P$ be a finite set of primes with $\lvert P\rvert \geq 2$ and let ${a_1<a_2<\cdots}={ n\in \mathbb{N} : \textrm{if }p\mid n\textrm{ then }p\in P}$. Is the sum

$$\sum_{n=1}^\infty \frac{1}{[a_1,\ldots,a_n]},$$

where $[a_1,\ldots,a_n]$ is the lowest common multiple of $a_1,\ldots,a_n$, irrational?

Status. Open.

Source. erdosproblems.com/269, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #269, https://www.erdosproblems.com/269.

References.

• [Er88c] Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109.

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

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Linked library material

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• erdos_1988_irrationality_certain_series_problems_results