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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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33
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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 (ghodges14(AT)comcast.net), 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
| 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.
R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 112.
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
| 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
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 (noe(AT)sspectra.com), Oct 15 2009]
Eric Weisstein's World of Mathematics, Superabundant Number
Wikipedia, Superabundant number
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MATHEMATICA
| a=0; Do[b=DivisorSigma[1, n]/n; If[b>a, a=b; Print[n]], {n, 1, 10^7}]
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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
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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 (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
| nonn,nice
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AUTHOR
| Matthew Conroy
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EXTENSIONS
| Matthew Conroy points out that these are different from the highly composite numbers - see A002182. Jul 10 1996.
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