A068403 Numbers k such that sigma(k) > 3*k.

180, 240, 360, 420, 480, 504, 540, 600, 660, 720, 780, 840, 900, 960, 1008, 1080, 1200, 1260, 1320, 1344, 1440, 1512, 1560, 1584, 1620, 1680, 1800, 1848, 1872, 1890, 1920, 1980, 2016, 2040, 2100, 2160, 2184, 2280, 2340, 2352, 2376, 2400, 2520, 2640

Offset: 1

Author

Benoit Cloitre, Mar 02 2002

Keywords

• easy
• nonn

Comments

Davenport shows that these numbers have positive density. Are there good bounds for the density?

G. Miller & M. Whalen suggested that 1018976683725 (3^3*5^2*7^2*11*13*17*19*23*29) might be the smallest odd number in the sequence (a fact now, see A119240 and A023197). - Michel Marcus, May 01 2013

From Amiram Eldar, Feb 13 2021: (Start)

Behrend (1933) found the bounds (0.009, 0.110) for the asymptotic density.

Wall et al. (1972) found the bounds (0.0186, 0.0461).

The upper bound was reduced to 0.0214614 using Deléglise's method by McDaniel College (2010). (End)

Note that 1018976683725, the smallest odd term in this sequence, is A053624(51). - Charles R Greathouse IV, Jan 09 2025

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

Terms not divisible by 6 are in A126104.

Cf. A001221, A068404, A215264.

Cf. A005820 (3-perfect numbers).

Programs

Maple

A068403:=n->`if`((numtheory)[sigma](n) > 3*n, n, NULL): seq(A068403(n), n=1..5*10^3); # Wesley Ivan Hurt, Apr 09 2017

Mathematica

Select[Range[180, 2000], 3*# < Plus@@Divisors[ # ]&] (* Vladimir Joseph Stephan Orlovsky, Apr 21 2010 *)
Select[Range[3000],DivisorSigma[1,#]>3#&] (* Harvey P. Dale, Aug 12 2023 *)

PARI

for(n=1, 3000, if(sigma(n)>3*n, print1(n,", "))) \\ Indranil Ghosh, Apr 10 2017

Python

from sympy import divisor_sigma
print([n for n in range(180, 3001) if divisor_sigma(n)>3*n]) # Indranil Ghosh, Apr 10 2017

Formula

A001221(a(n)) >= 3 (Laatsch, 1986). - Amiram Eldar, Nov 07 2020

a(n) ~ k*n for some constant k with 46 < k < 54. - Charles R Greathouse IV, Jan 21 2025