

A004394


Superabundant [or superabundant] numbers: n such that sigma(n)/n > sigma(m)/m for all m<n, sigma(n) being the sum of the divisors of n.


38



1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 332640, 554400, 665280, 720720, 1441440, 2162160, 3603600, 4324320, 7207200, 8648640, 10810800, 21621600
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OFFSET

1,2


COMMENTS

Also n such that sigma_{1}(n) > sigma_{1}(m) for all m < n, where sigma_{1}(n) is the sum of the reciprocals of the divisors of n.  Matthew Vandermast, Jun 09 2004
Ramanujan (1997, Section 59; written in 1915) called these numbers "generalized highly composite." Alaoglu and Erdos (1944) changed the terminology to "superabundant." [Jonathan Sondow, Jul 11 2011]
Alaoglu and Erdos show that: (1) n is superabundant => n=2^{e_2} * 3^{e_3} * ...* p^{e_p}, with e_2>=e_3>=...>=e_p (and e_p is 1 unless n=4 or n=36); (2) if q<r are primes, then  e_r  floor(e_q*log(q)/log(r))  <= 1; (3) q^{e_q}<2^{e_2+2} for primes q, 2<q<=p.  Keith Briggs (keith.briggs(AT)bt.com), Apr 26 2005
See A166735 for superabundant numbers that are not highly composite, and A189228 for superabundant numbers that are not colossally abundant.


REFERENCES

R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 112.
J. Sandor, "Abundant numbers", In: M. Hazewinkel, Encyclopedia of Mathematics, Supplement III, Kluwer Acad.Publ., 2002 (see pp. 1921).
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 128.


LINKS

T. D. Noe and D. Kilminster, Table of n, a(n) for n=0..2000 (First 500 terms from T. D. Noe. Extends to n=8436 in the comments.)
A. Akbary and Z. Friggstad, Superabundant numbers and the Riemann hypothesis, Amer. Math. Monthly, 116 (2009), 273275.
L. Alaoglu and P. Erdos, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448469. Errata
Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 16 (2006), p. 251256.
Geoffrey Caveney, JeanLouis Nicolas and Jonathan Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, INTEGERS 11 (2011), #A33.
G. Caveney, J.L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Arxiv preprint arXiv:1112.6010, 2011.  From N. J. A. Sloane, Apr 14 2012
G. Caveney, J.L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359384.
Matthew M. Conroy, Home page (listed instead of email address)
P. Erdos & J.L. Nicolas, Repartition des nombres superabondants (Text in French)
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534543.
A. Morkotun, On the increase of Gronwall function value at the multiplication of its argument by a prime, arXiv preprint arXiv:1307.0083, 2013
S. Nazardonyavi and S. Yakubovich, Superabundant numbers, their subsequences and the Riemann hypothesis, arXiv preprint arXiv:1211.2147, 2012.
S. Nazardonyavi and S. Yakubovich, Delicacy of the Riemann hypothesis and certain subsequences of superabundant numbers, arXiv preprint arXiv:1306.3434, 2013
S. Nazardonyavi, S. Yakubovich, Extremely Abundant Numbers and the Riemann Hypothesis, Journal of Integer Sequences, 17 (2014), Article 14.2.8.
Walter Nissen, Abundancy : Some Resources
T. D. Noe, First 500 superabundant numbers
T. D. Noe, First 1000000 superabundant numbers (21 MB, zipped)
S. Ramanujan, Highly composite numbers, Annotated and with a foreword by J.L. Nicholas and G. Robin, Ramanujan J., 1 (1997), 119153.
T. SchwabhÃ¤user, Preventing Exceptions to Robin's Inequality, arXiv preprint arXiv:1308.3678, 2013
Eric Weisstein's World of Mathematics, Superabundant Number
Wikipedia, Superabundant number


MATHEMATICA

a=0; Do[b=DivisorSigma[1, n]/n; If[b>a, a=b; Print[n]], {n, 1, 10^7}]


PROG

(PARI) print1(r=1); forstep(n=2, 1e6, 2, t=sigma(n, 1); if(t>r, r=t; print1(", "n))) \\ Charles R Greathouse IV, Jul 19 2011


CROSSREFS

Cf. A002182, A002093; colossally abundant numbers: A004490.
A023199 is a subsequence. Almost same as A077006.
Cf. A112974 (number of superabundant numbers between colossally abundant numbers).
Cf. A091901 (Robin's inequality), A189686 (superabundant and the reverse of Robin's inequality), A192884 (nonsuperabundant and the reverse of Robin's inequality).
Sequence in context: A002182 A077006 A166981 * A137425 A141320 A134865
Adjacent sequences: A004391 A004392 A004393 * A004395 A004396 A004397


KEYWORD

nonn,nice


AUTHOR

Matthew Conroy


EXTENSIONS

Matthew Conroy points out that these are different from the highly composite numbers  see A002182. Jul 10 1996


STATUS

approved



