New proof and generalization of some results on translated sums over k-almost primes
A sequence $\mathcal{A}$ of strictly positive integers is said to be primitive if none of its terms divides the others, Erdős conjectured that the sum $f(\mathcal{A},0)\le f(\mathbb{N}_{1},0),$ where $\mathbb{N}_{1}$ is the sequence of prime numbers and $f(\mathcal{A},h)=\sum _{a\,\in \,\mathcal{A}}...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2024-05-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.552/ |
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| Summary: | A sequence $\mathcal{A}$ of strictly positive integers is said to be primitive if none of its terms divides the others, Erdős conjectured that the sum $f(\mathcal{A},0)\le f(\mathbb{N}_{1},0),$ where $\mathbb{N}_{1}$ is the sequence of prime numbers and $f(\mathcal{A},h)=\sum _{a\,\in \,\mathcal{A}}$ $\frac{1}{a(\log a+h) }$. In 2019, Laib et al. proved that the analogous conjecture of Erdős $f(\mathcal{A},h)\le f(\mathbb{N}_{1},h)$ is false for $h\ge 81$ on a sequence of semiprimes. Recently, Lichtman gave the best lower bound $h=1.04\cdots $ on semiprimes and he obtained other results for translated sums on $k$-almost primes with $ 2 |
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| ISSN: | 1778-3569 |