A068403 - OEIS (original) (raw)

A068403

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

27

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

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

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)

Odd terms have a prime factor p >= 23. 75369137468625 = 3^4 * 5^3 * 7^2 * 11^2 * 13^2 * 17 * 19 * 23 is the first odd term whose greatest prime factor is 23. - Peter Munn, Sep 24 2025

REFERENCES

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

LINKS

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.

MATHEMATICA

Select[Range[3000], DivisorSigma[1, #]>3#&] (* Harvey P. Dale, Aug 12 2023 *)

PROG

(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

CROSSREFS

Terms not divisible by 6 are in A126104.

Cf. A005820 (3-perfect numbers).