Search: id:A004394
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%I A004394
%S A004394 1,2,4,6,12,24,36,48,60,120,180,240,360,720,840,1260,1680,2520,5040,
%T A004394 10080,15120,25200,27720,55440,110880,166320,277200,332640,554400,665280,
%U A004394 720720,1441440,2162160,3603600,4324320,7207200,8648640,10810800
%N A004394 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.
%C A004394 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
%C A004394 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
%D A004394 L. Alaoglu and P. Erdos, On highly composite and similar numbers, Trans. 
               Amer. Math. Soc., 56 (1944), 448-469.
%D A004394 A. Akbary and Z. Friggstad, Superabundant numbers and the Riemann hypothesis, 
               Amer. Math. Monthly, 116 (2009), 273-275.
%D A004394 R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 112.
%D A004394 J. Sandor, "Abundant numbers", In: M. Hazewinkel, Encyclopedia of Mathematics, 
               Supplement III, Kluwer Acad.Publ., 2002 (see pp. 19-21).
%D A004394 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. 
               Penguin Books, NY, 1986, 128.
%H A004394 D. Kilminster, <a href="b004394.txt">Table of n, a(n) for n=0..2000</
               a> (Extends to n=8436 in the comments.)
%H A004394 Matthew M. Conroy, <a href="http://www.madandmoonly.com/doctormatt">Home 
               page</a> (listed instead of email address)
%H A004394 P. Erdos & J.-L. Nicolas, <a href="http://archive.numdam.org/article/
               BSMF_1975__103__65_0.pdf">Repartition des nombres superabondants 
               (Text in French)</a>
%H A004394 J. C. Lagarias, <a href="http://arXiv.org/abs/math.NT/0008177">An elementary 
               problem equivalent to the Riemann hypothesis</a>, Am. Math. Monthly 
               109 (#6, 2002), 534-543.
%H A004394 Walter Nissen, <a href="http://upforthecount.com/math/abundance.html">
               Abundancy : Some Resources </a>
%H A004394 T. D. Noe, <a href="http://www.sspectra.com/math/SAN.txt">First 500 superabundant 
               numbers</a>
%H A004394 T. D. Noe, <a href="http://www.sspectra.com/math/SAN_1000000.zip">First 
               1000000 superabundant numbers (21 MB, zipped)</a> [From T. D. Noe 
               (noe(AT)sspectra.com), Oct 15 2009]
%H A004394 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SuperabundantNumber.html">Superabundant Number</a>
%H A004394 Wikipedia, <a href="http://en.wikipedia.org/wiki/Superabundant_number">
               Superabundant number</a>
%t A004394 a=0; Do[b=DivisorSigma[1, n]/n; If[b>a, a=b; Print[n]], {n, 1, 10^7}]
%Y A004394 Cf. A002182, A002093; colossally abundant numbers: A004490.
%Y A004394 A023199 is a subsequence. Almost same as A077006.
%Y A004394 Cf. A112974 (number of superabundant numbers between colossally abundant 
               numbers).
%Y A004394 Sequence in context: A094348 A002182 A077006 this_sequence A166981 A137425 
               A141320
%Y A004394 Adjacent sequences: A004391 A004392 A004393 this_sequence A004395 A004396 
               A004397
%K A004394 nonn,nice
%O A004394 1,2
%A A004394 Matthew Conroy
%E A004394 Matthew Conroy points out that these are different from the highly composite 
               numbers - see A002182. Jul 10 1996.

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