Combinatoire, Théorie des nombres
Translated sums of primitive sets
Lichtman, Jared Duker1
1 Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK
Comptes Rendus. Mathématique, Tome 360 (2022) no. G4, pp. 409-414.

The Erdős primitive set conjecture states that the sum f(A)=∑a∈A1aloga, ranging over any primitive set A of positive integers, is maximized by the set of prime numbers. Recently Laib, Derbal, and Mechik proved that the translated Erdős conjecture for the sum f(A,h)=∑a∈A1a(loga+h) is false starting at h=81, by comparison with semiprimes. In this note we prove that such falsehood occurs already at h=1.04⋯, and show this translate is best possible for semiprimes. We also obtain results for translated sums of k-almost primes with larger k.

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DOI : 10.5802/crmath.285
Classification : 11N25, 11Y60, 11A05, 11M32
Lichtman, Jared Duker 1

1 Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK 
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Lichtman, Jared Duker. Translated sums of primitive sets. Comptes Rendus. Mathématique, Tome 360 (2022) no. G4, pp. 409-414. doi : 10.5802/crmath.285. https://www.numdam.org/articles/10.5802/crmath.285/

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  • Laib, Ilias Translated sums on special family of quasi-primitive sequences, International Journal of Number Theory, Volume 21 (2025) no. 03, p. 615 | DOI:10.1142/s1793042125500307
  • Gorodetsky, Ofir; Lichtman, Jared Duker; Wong, Mo Dick On Erdős sums of almost primes, Comptes Rendus. Mathématique, Volume 362 (2024) no. G12, p. 1571 | DOI:10.5802/crmath.650
  • Laib, Ilias New proof and generalization of some results on translated sums over k-almost primes, Comptes Rendus. Mathématique, Volume 362 (2024) no. G5, p. 481 | DOI:10.5802/crmath.552
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  • Lichtman, Jared Duker A proof of the Erdős primitive set conjecture, Forum of Mathematics, Pi, Volume 11 (2023) | DOI:10.1017/fmp.2023.16

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