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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}. 34

%I

%S 2,6,12,60,120,360,2520,5040,55440,720720,1441440,4324320,21621600,

%T 367567200,6983776800,160626866400,321253732800,9316358251200,

%U 288807105787200,2021649740510400,6064949221531200,224403121196654400

%N 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}.

%D 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.

%H T. D. Noe, <a href="/A004490/b004490.txt">Table of n, a(n) for n = 1..150</a>

%H L. Alaoglu and P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1944-03.pdf">On highly composite and similar numbers,</a> Trans. Amer. Math. Soc., 56 (1944), 448-469. <a href="http://upforthecount.com/math/errata.html">Errata</a>

%H G. Caveney, J.-L. Nicolas and J. Sondow, <a href="http://arxiv.org/abs/1112.6010">On SA, CA, and GA numbers</a>, Ramanujan J., 29 (2012), 359-384.

%H Keith Briggs, <a href="https://projecteuclid.org/euclid.em/1175789744">Abundant numbers and the Riemann Hypothesis</a>, Experimental Math., Vol. 16 (2006), p. 251-256.

%H G. Caveney, J.-L. Nicolas and J. Sondow, <a href="http://arxiv.org/abs/1112.6010">On SA, CA, and GA numbers</a>, arXiv preprint arXiv:1112.6010 [math.NT], 2011. - From _N. J. A. Sloane_, Apr 14 2012

%H 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 S. Nazardonyavi, S. Yakubovich, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Nazar/nazar4.html">Extremely Abundant Numbers and the Riemann Hypothesis</a>, Journal of Integer Sequences, 17 (2014), Article 14.2.8.

%H S. Ramanujan, <a href="http://dx.doi.org/10.1023/A:1009764017495">Highly composite numbers</a>, Annotated and with a foreword by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119-153.

%H T. Schwabhäuser, <a href="http://arxiv.org/abs/1308.3678">Preventing Exceptions to Robin's Inequality</a>, arXiv preprint arXiv:1308.3678 [math.NT], 2013.

%H M. Waldschmidt, <a href="http://www.math.jussieu.fr/~miw/articles/pdf/LegacyRamanujan2013Text.pdf">From highly composite numbers to transcendental number theory</a>, 2013.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ColossallyAbundantNumber.html">Colossally Abundant Number</a>

%Y A subset of A004394. Cf. A002201.

%Y Cf. A073751.

%Y Cf. abundant numbers = A002093, A002182, A005101, A006038, A004394; highly abundant numbers = A002093, superabundant numbers = A004394, superabundant numbers that are not colossally abundant = A189228.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Jan 22 2001

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Last modified October 16 18:03 EDT 2019. Contains 328102 sequences. (Running on oeis4.)