A302991 Decimal expansion of the asymptotic density of abundant numbers.

2, 4, 7, 6, 1, 9, 6

Offset: 0

Author

Amiram Eldar, Apr 17 2018

Keywords

• nonn
• cons
• more

Comments

The existence of this density was asked about by Erich Bessel-Hagen in 1929 and was proved by Harold Davenport in 1933.

Previous evaluated bounds on the value of density, D, were D < 0.47 (Behrend, 1932), 0.241 < D < 0.314 (Behrend, 1933), 0.24432 < D (Salié, 1955), 0.2441 < D < 0.2909 (Wall et al., 1972), and 0.2474 < D < 0.2480 (Deléglise, 1988); the current bounds are 0.2476171 < D < 0.2476475 (Kobayashi, 2010).

0.247619608 < D < 0.247619658 (McNew and Setty, 2025). - Amiram Eldar, Aug 02 2025

Examples

			0.2476196...
		

References

• Felix Behrend, Über numeri abundantes I, S.-Ber. Preuß. Akad. Wiss., math.-nat. Kl. (1932), pp. 322-328.
• Erich Bessel-Hagen, Repertorium der höheren Mathematik, 2nd edn., Vol. 1, B. G. Teubner (Leipzig, 1929), pp. 1458-1574.
• Harold Davenport, Über numeri abundantes, Sitzungsberichte der Preußischen Akademie der Wissenschaften, phys.-math. Klasse, No. 6 (1933), pp. 830-837.

Links

• Felix Behrend, Über numeri abundantes, inaugural dissertation, Sitzungsberichte der Preußischen Akademie der Wissenschaften, phys.-math. Klasse, pp. 280-293, 1933.
• Marc Deléglise, Bounds for the Density of Abundant Integers, Experimental Mathematics, Vol. 7, No. 2 (1998), pp. 137-143.
• Mitsuo Kobayashi, On the Density of Abundant Numbers, Ph.D. thesis, Dartmouth College, 2010.
• Nathan McNew and Jai Setty, On the densities of covering numbers and abundant numbers, arXiv:2507.23041 [math.NT], 2025.
• Hans Salié, Über die Dichte abundanter Zahlen, Mathematische Nachrichten, Vol. 14, No. 1 (1955), pp. 39-46.
• Charles R. Wall, Phillip L. Crews and Donald B. Johnson, Density Bounds for the Sum of Divisors Function, Mathematics of Computation, Vol. 26, No. 119 (1972), pp. 773-777; Errata, Vol. 31, No. 138 (1977), p. 616.

Crossrefs

Cf. A000203, A005101, A302992.

Extensions

a(4)-a(5) from Muniru A Asiru, Aug 20 2018

a(6) from Amiram Eldar, Aug 02 2025