A068404 - OEIS

%I #36 Feb 13 2021 05:56:28

%S 27720,50400,55440,60480,65520,75600,83160,85680,90720,95760,98280,

%T 100800,105840,110880,115920,120120,120960,128520,131040,138600,

%U 141120,143640,151200,163800,166320,171360,176400,180180,181440,184800,191520

%N Numbers k such that sigma(k) > 4*k.

%C This sequence is of positive density, see for example Davenport. The density is between 0.000176 and 0.004521 according to the McDaniel College link. - _Charles R Greathouse IV_, Sep 07 2012

%C From _Amiram Eldar_, Feb 13 2021: (Start)

%C Behrend (1933) found the bounds (0.00003, 0.025) for the asymptotic density.

%C Wall et al. (1972) found the bounds (0.0001, 0.0147).

%C Using Deléglise's method the upper bound for the density found by McDaniel College is 0.000679406. (End)

%D Harold Davenport, Über numeri abundantes, Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl., No. 6 (1933), pp. 830-837.

%H Amiram Eldar, <a href="/A068404/b068404.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)

%H Felix Behrend, <a href="https://eudml.org/doc/204583">Über numeri abundantes II</a>, Preuss. Akad. Wiss. Sitzungsber., Vol. 6 (1933), pp. 280-293; <a href="http://mcdanielabundancy.wdfiles.com/local--files/bounds-for-abundancy-density/Behrend.pdf">alternative link</a>.

%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 Richard Laatsch, <a href="http://www.jstor.org/stable/2690424">Measuring the Abundancy of Integers</a>, Mathematics Magazine, Vol. 59, No. 2 (1986), pp. 84-92, <a href="https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/99C5C5IC/Laatsch%20-%201986%20-%20Measuring%20the%20Abundancy%20of%20Integers.pdf">alternative link</a>.

%H Gordon L. Miller and Mary T. Whalen, <a href="https://doi.org/10.1111/j.1949-8594.1995.tb15776.x">Multiply Abundant Numbers</a>, School Science and Mathematics, Volume 95, Issue 5 (May 1995), pp. 256-259.

%H Summer 2010 research group on Abundancy, <a href="http://mcdanielabundancy.wikidot.com/result-page">Abundancy Bounds 2010</a>, McDaniel College, 2010.

%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; <a href="https://doi.org/10.1090/S0025-5718-1977-0427251-X">Errata</a>, Vol. 31, No. 138 (1977), p. 616.

%F A001221(a(n)) >= 4 (Laatsch, 1986). - _Amiram Eldar_, Nov 07 2020

%t Select[Range[27720,9!,60], 4*#<Plus@@Divisors[ # ]&] (* _Vladimir Joseph Stephan Orlovsky_, Apr 21 2010 *)

%Y Cf. A068403, A001221, A215264.

%Y Cf. A027687 (4-perfect numbers).

%K nonn

%O 1,1

%A _Benoit Cloitre_, Mar 02 2002