cp's OEIS Frontend

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

%I A068403 #72 Jan 21 2025 13:31:16
%S A068403 180,240,360,420,480,504,540,600,660,720,780,840,900,960,1008,1080,
%T A068403 1200,1260,1320,1344,1440,1512,1560,1584,1620,1680,1800,1848,1872,
%U A068403 1890,1920,1980,2016,2040,2100,2160,2184,2280,2340,2352,2376,2400,2520,2640
%N A068403 Numbers k such that sigma(k) > 3*k.
%C A068403 Davenport shows that these numbers have positive density. Are there good bounds for the density?
%C A068403 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
%C A068403 From _Amiram Eldar_, Feb 13 2021: (Start)
%C A068403 Behrend (1933) found the bounds (0.009, 0.110) for the asymptotic density.
%C A068403 Wall et al. (1972) found the bounds (0.0186, 0.0461).
%C A068403 The upper bound was reduced to 0.0214614 using Deléglise's method by McDaniel College (2010). (End)
%C A068403 Note that 1018976683725, the smallest odd term in this sequence, is A053624(51). - _Charles R Greathouse IV_, Jan 09 2025
%D A068403 Harold Davenport, Über numeri abundantes, Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl., No. 6 (1933), pp. 830-837.
%H A068403 Amiram Eldar, <a href="/A068403/b068403.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H A068403 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 A068403 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 A068403 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 A068403 Gordon L. Miller and Mary T. Whalen, <a href="https://dx.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 A068403 Summer 2010 research group on Abundancy, <a href="http://mcdanielabundancy.wikidot.com/result-page">Abundancy Bounds 2010</a>, McDaniel College, 2010.
%H A068403 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 A068403 A001221(a(n)) >= 3 (Laatsch, 1986). - _Amiram Eldar_, Nov 07 2020
%F A068403 a(n) ~ k*n for some constant k with 46 < k < 54. - _Charles R Greathouse IV_, Jan 21 2025
%p A068403 A068403:=n->`if`((numtheory)[sigma](n) > 3*n, n, NULL): seq(A068403(n), n=1..5*10^3); # _Wesley Ivan Hurt_, Apr 09 2017
%t A068403 Select[Range[180, 2000], 3*# < Plus@@Divisors[ # ]&] (* _Vladimir Joseph Stephan Orlovsky_, Apr 21 2010 *)
%t A068403 Select[Range[3000],DivisorSigma[1,#]>3#&] (* _Harvey P. Dale_, Aug 12 2023 *)
%o A068403 (PARI) for(n=1, 3000, if(sigma(n)>3*n, print1(n,", "))) \\ _Indranil Ghosh_, Apr 10 2017
%o A068403 (Python)
%o A068403 from sympy import divisor_sigma
%o A068403 print([n for n in range(180, 3001) if divisor_sigma(n)>3*n]) # _Indranil Ghosh_, Apr 10 2017
%Y A068403 Terms not divisible by 6 are in A126104.
%Y A068403 Cf. A001221, A068404, A215264.
%Y A068403 Cf. A005820 (3-perfect numbers).
%K A068403 easy,nonn
%O A068403 1,1
%A A068403 _Benoit Cloitre_, Mar 02 2002