A068404 Numbers k such that sigma(k) > 4*k.

27720, 50400, 55440, 60480, 65520, 75600, 83160, 85680, 90720, 95760, 98280, 100800, 105840, 110880, 115920, 120120, 120960, 128520, 131040, 138600, 141120, 143640, 151200, 163800, 166320, 171360, 176400, 180180, 181440, 184800, 191520

Offset: 1

Author

Benoit Cloitre, Mar 02 2002

Keywords

• nonn

Comments

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

From Amiram Eldar, Feb 13 2021: (Start)

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

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

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

References

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Felix Behrend, Über numeri abundantes II, Preuss. Akad. Wiss. Sitzungsber., Vol. 6 (1933), pp. 280-293; alternative link.
• Marc Deléglise, Bounds for the Density of Abundant Integers, Experimental Mathematics, Vol. 7, No. 2 (1998), pp. 137-143.
• Richard Laatsch, Measuring the Abundancy of Integers, Mathematics Magazine, Vol. 59, No. 2 (1986), pp. 84-92, alternative link.
• Gordon L. Miller and Mary T. Whalen, Multiply Abundant Numbers, School Science and Mathematics, Volume 95, Issue 5 (May 1995), pp. 256-259.
• Summer 2010 research group on Abundancy, Abundancy Bounds 2010, McDaniel College, 2010.
• 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. A068403, A001221, A215264.

Cf. A027687 (4-perfect numbers).

Programs

Mathematica

Select[Range[27720,9!,60], 4*#Vladimir Joseph Stephan Orlovsky, Apr 21 2010 *)

Formula

A001221(a(n)) >= 4 (Laatsch, 1986). - Amiram Eldar, Nov 07 2020