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A068404
Numbers k such that sigma(k) > 4*k.
9
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
1,1
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.
FORMULA
A001221(a(n)) >= 4 (Laatsch, 1986). - Amiram Eldar, Nov 07 2020
MATHEMATICA
Select[Range[27720, 9!, 60], 4*#<Plus@@Divisors[ # ]&] (* Vladimir Joseph Stephan Orlovsky, Apr 21 2010 *)
CROSSREFS
Cf. A027687 (4-perfect numbers).
Sequence in context: A204831 A345153 A190111 * A279091 A307114 A291458
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Mar 02 2002
STATUS
approved