Paul Erd{o}s asked how rapidly a sequence of positive integers (nₖ) can grow if both series ∑ₖ 1/nₖ and ∑ₖ 1/(nₖ-1) have rational sums. In this note we show that there exists an exponentially growing sequence (nₖ) with this property. Previous records had polynomial growth, even for easier variants of the problem, regarding the series ∑ₖ 1/nₖ and ∑ₖ 1/(nₖ-d) for any concrete nonzero integer d.
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Vjekoslav Kovač (2024) studied this question.
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