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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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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| author | Laib, Ilias |
| author_facet | Laib, Ilias |
| author_sort | Laib, Ilias |
| collection | DOAJ |
| description | 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 |
| format | Article |
| id | doaj-art-00c2ae8e581b4f8f9381a9ec34e2dfbc |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-05-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-00c2ae8e581b4f8f9381a9ec34e2dfbc2025-02-07T11:21:12ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-05-01362G548148610.5802/crmath.55210.5802/crmath.552New proof and generalization of some results on translated sums over k-almost primesLaib, Ilias0ENSTP and Laboratory of Equations with Partial Non-Linear Derivatives ENS Vieux Kouba, Algiers, Algeria.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 $ 2https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.552/ |
| spellingShingle | Laib, Ilias New proof and generalization of some results on translated sums over k-almost primes Comptes Rendus. Mathématique |
| title | New proof and generalization of some results on translated sums over k-almost primes |
| title_full | New proof and generalization of some results on translated sums over k-almost primes |
| title_fullStr | New proof and generalization of some results on translated sums over k-almost primes |
| title_full_unstemmed | New proof and generalization of some results on translated sums over k-almost primes |
| title_short | New proof and generalization of some results on translated sums over k-almost primes |
| title_sort | new proof and generalization of some results on translated sums over k almost primes |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.552/ |
| work_keys_str_mv | AT laibilias newproofandgeneralizationofsomeresultsontranslatedsumsoverkalmostprimes |