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A004394
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Superabundant [or super-abundant] numbers: n such that sigma(n)/n > sigma(m)/m for all m<n, sigma(n) being the sum of the divisors of n.
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36
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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
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OFFSET
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1,2
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COMMENTS
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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.
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REFERENCES
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A. Akbary and Z. Friggstad, Superabundant numbers and the Riemann hypothesis, Amer. Math. Monthly, 116 (2009), 273-275.
Geoffrey Caveney, Jean-Louis 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
R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 112.
S. Nazardonyavi and S. Yakubovich, Superabundant numbers, their subsequences and the Riemann hypothesis, arXiv preprint arXiv:1211.2147, 2012. - From N. J. A. Sloane, Jan 02 2013
S. Ramanujan, Highly composite numbers, Annotated and with a foreword by J.-L. Nicholas and G. Robin, Ramanujan J., 1 (1997), 119-153.
J. Sandor, "Abundant numbers", In: M. Hazewinkel, Encyclopedia of Mathematics, Supplement III, Kluwer Acad.Publ., 2002 (see pp. 19-21).
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 128.
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LINKS
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D. Kilminster, Table of n, a(n) for n=0..2000 (Extends to n=8436 in the comments.)
L. Alaoglu and P. Erdos, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata
G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384.
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), 534-543.
Walter Nissen, Abundancy : Some Resources
T. D. Noe, First 500 superabundant numbers
T. D. Noe, First 1000000 superabundant numbers (21 MB, zipped) [From T. D. Noe, Oct 15 2009]
Eric Weisstein's World of Mathematics, Superabundant Number
Wikipedia, Superabundant number
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MATHEMATICA
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a=0; Do[b=DivisorSigma[1, n]/n; If[b>a, a=b; Print[n]], {n, 1, 10^7}]
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PROG
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(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
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CROSSREFS
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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 (non-superabundant and the reverse of Robin's inequality).
Sequence in context: A002182 A077006 * A166981 A137425 A141320 A134865
Adjacent sequences: A004391 A004392 A004393 * A004395 A004396 A004397
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KEYWORD
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nonn,nice,changed
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AUTHOR
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Matthew Conroy
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EXTENSIONS
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Matthew Conroy points out that these are different from the highly composite numbers - see A002182. Jul 10 1996.
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STATUS
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approved
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