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Decimal expansion of the asymptotic density of abundant numbers.
13

%I #21 Oct 06 2018 11:47:00

%S 2,4,7,6,1,9

%N Decimal expansion of the asymptotic density of abundant numbers.

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

%C 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).

%D Felix Behrend, Über numeri abundantes I, S.-Ber. Preuß. Akad. Wiss., math.-nat. Kl. (1932), pp. 322-328.

%D Erich Bessel-Hagen, Repertorium der höheren Mathematik, 2nd edn., Vol. 1, B. G. Teubner (Leipzig, 1929), pp. 1458-1574.

%D Harold Davenport, Über numeri abundantes, Sitzungsberichte der Preußischen Akademie der Wissenschaften, phys.-math. Klasse, No. 6 (1933), pp. 830-837.

%H Felix Behrend, <a href="https://gdz.sub.uni-goettingen.de/id/PPN313148708">Über numeri abundantes</a>, inaugural dissertation, Sitzungsberichte der Preußischen Akademie der Wissenschaften, phys.-math. Klasse, pp. 280-293, 1933.

%H Marc Deléglise, <a href="https://doi.org/10.1080/10586458.1998.10504363">Bounds for the Density of Abundant Integers</a>, Experimental Mathematics, Vol. 7, No. 2 (1998), pp. 137-143.

%H Mitsuo Kobayashi, <a href="http://dx.doi.org/10.1349/ddlp.1662">On the Density of Abundant Numbers</a>, Ph.D. thesis, Dartmouth College, 2010.

%H H. Salié, <a href="https://doi.org/10.1002/mana.19550140107">Über die Dichte abundanter Zahlen</a>, Mathematische Nachrichten, Vol. 14, No. 1 (1955), pp. 39-46.

%H Charles R. Wall, Phillip L. Crews and Donald B. Johnson, <a href="https://doi.org/10.1090/S0025-5718-1972-0327700-7 ">Density Bounds for the Sum of Divisors Function</a>, Mathematics of Computation, Vol. 26, No. 119 (1972), pp. 773-777; Errata, Vol. 31, No. 138 (1977), p. 616.

%e 0.2476...

%Y Cf. A000203, A005101, A302992.

%K nonn,cons,more

%O 0,1

%A _Amiram Eldar_, Apr 17 2018

%E a(4)-a(5) from _Muniru A Asiru_, Aug 20 2018