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On the solvability of the equations \([a_i,a_j] = a_r\) and \((a'_i,a'_j) = a'_r\) in sequences of positive density. (English) Zbl 0151.03502

The authors obtain the following results.
1) Let \(a_1<a_2<\cdots\) be an infinite sequence of integers for which there are infinitely many integers \(n_1 < n_2 < \cdots\) satisfying \[ \sum_{a_i<n_k} {1 \over a_i} > c_1 {\log n_k \over (\log\log n_k)^{1/2}}. \] Then the equations \((a'_i,a'_j)=a'_r,[a_i,a_j] = a_r\) have infinitely many solutions. The symbol \((a_i,a_j)\) denotes the greatest common divisor and \([a_i,a_j]\) denotes the least common multiple of \(a_i\) and \(a_j\).
2) Let \(a_1 < a_2 < \cdots\) be an infinite sequence of integers for which there are infinitely many integers \(n_1 < n_2 < \cdots\) satisfying \[ \sum_{a_i<n_k} {1 \over a_i}>c_2 {\log n_k \over (\log\log n_k)^{1/4}}. \] Then there are infinitely many quadruplets of distinct integers \(a_i,a_j,a_r,a_s\) satisfying \((a_i,a_j) = a_r,\) \([a_i,a_j] = a_s\), where \(c_1\) and \(c_2\) denote suitable positive constants.
Reviewer: Cs. Pogany

MSC:

11B83 Special sequences and polynomials
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References:

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