Deléglise, Marc Bounds for the density of abundant integers. (English) Zbl 0923.11127 Exp. Math. 7, No. 2, 137-143 (1998). An integer \(n\) is called abundant if the sum of the divisors of \(n\) is at least \(2n\). It is known that abundant numbers have natural density \(d\) and that \(0.244<d<0.291\). The author answers a question by H. Cohen, showing that \(d<1/4\). Precisely, using a method of Behrend the authors shows that \(0.2474<d<0.2480\). Reviewer: Alberto Perelli (Genova) Cited in 1 ReviewCited in 9 Documents MSC: 11N25 Distribution of integers with specified multiplicative constraints 11A25 Arithmetic functions; related numbers; inversion formulas Keywords:density of abundant integers; sum of divisors; abundant numbers PDF BibTeX XML Cite \textit{M. Deléglise}, Exp. Math. 7, No. 2, 137--143 (1998; Zbl 0923.11127) Full Text: DOI Euclid EuDML EMIS Online Encyclopedia of Integer Sequences: Deficient numbers: numbers k such that sigma(k) < 2k. Abundance of n, or (sum of divisors of n) - 2n. Numbers k such that sigma(k) > 3*k. Numbers k such that sigma(k) > 4*k. Number of abundant numbers <= n. Number of deficient numbers <= n. Decimal expansion of the asymptotic density of abundant numbers. References: [1] Behrend F., Sitzungsber. Preuss. Akad. Wiss. pp 280– (1933) [2] Davenport H., Sitzungsber. Preuss. Akad. Wiss. pp 830– (1933) [3] Del M., J. Théor. Nombres Bordeaux 6 (2) pp 327– (1994) · Zbl 0839.11041 [4] Elliott P. D. T. A., Probabilistic number theory, I: Mean-value theorems (1979) · Zbl 0431.10029 [5] Glaisher W. L., Quaterly Journal of Math. 25 pp 347– (1891) [6] Martinet J., Séminaire de Théorie des Nombres (1973) [7] Tenenbaum G., Introduction à la théorie analytique et probabiliste des nombres,, 2. ed. (1995) · Zbl 0880.11001 [8] Wall C. R., The theory of arithmetic functions (Kalamazoo, Mich., 1971) pp 283– (1972) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.