%I A004490
%S A004490 2,6,12,60,120,360,2520,5040,55440,720720,1441440,4324320,21621600,
%T A004490 367567200,6983776800,160626866400,321253732800,9316358251200,288807105787200,
%U A004490 2021649740510400,6064949221531200,224403121196654400
%N A004490 Colossally abundant numbers: n for which there is a positive exponent 
               epsilon such that sigma(n)/n^{1 + epsilon} >= sigma(k)/k^{1 + epsilon} 
               for all k > 1, so that n attains the maximum value of sigma(n)/n^{1 
               + epsilon}.
%C A004490 All superabundant, colossally abundant, highly composite and superior 
               highly composite numbers are Niven/Harshad numbers. - Rob Hoogers 
               (chimera(AT)chimera.fol.nl), Jun 26 2004
%C A004490 The previous comment is erroneous. The first superabundant number that 
               is not a Harshad number is A004394(105) = 149602080797769600. The 
               first highly composite number that is not a Harshad number is A002182(61) 
               = 245044800. For all exceptions I found, the sum of digits is a power 
               of 3. Although the first 60000 terms of the colossally abundant numbers 
               and the superior highly composite numbers are Harshad numbers, I 
               am not aware of a proof that all terms are Harshad numbers. There 
               may be large counterexamples. [From T. D. Noe (noe(AT)sspectra.com), 
               Oct 27 2009]
%D A004490 L. Alaoglu and P. Erdos, On Highly Composite and Similar Numbers, Trans. 
               Amer. Math. Soc. 56 (1944), 448-469.
%D A004490 S. Ramanujan, Highly composite numbers, Proc. London Math. Soc., 14 (1915), 
               347-407. Reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 
               1927; Chelsea, NY, 1962, pp. 78-129. See esp. pp. 87, 115.
%D A004490 S. Ramanujan, Highly composite numbers, Annotated and with a foreword 
               by J.-L. Nicholas and G. Robin, Ramanujan J., 1 (1997), 119-153.
%H A004490 T. D. Noe, <a href="b004490.txt">Table of n, a(n) for n = 1..150</a>
%H A004490 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 A004490 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ColossallyAbundantNumber.html">Colossally Abundant Number</a>
%Y A004490 A subset of A004394. Cf. A002201.
%Y A004490 Cf. A073751.
%Y A004490 Cf. abundant numbers = A002093, A002182, A005101, A006038, A004394; colossally 
               abundant numbers = A004490, highly abundant numbers = A002093, superabundant 
               numbers = A004394.
%Y A004490 Sequence in context: A072181 A126915 A002201 this_sequence A135060 A072486 
               A096123
%Y A004490 Adjacent sequences: A004487 A004488 A004489 this_sequence A004491 A004492 
               A004493
%K A004490 nonn
%O A004490 1,1
%A A004490 N. J. A. Sloane (njas(AT)research.att.com), Jan 22 2001