cp's OEIS Frontend

A004490 Colossally abundant numbers: m for which there is a positive exponent epsilon such that sigma(m)/m^{1 + epsilon} >= sigma(k)/k^{1 + epsilon} for all k > 1, so that m attains the maximum value of sigma(m)/m^{1 + epsilon}.

%I A004490 #89 Aug 15 2025 11:29:26
%S A004490 2,6,12,60,120,360,2520,5040,55440,720720,1441440,4324320,21621600,
%T A004490 367567200,6983776800,160626866400,321253732800,9316358251200,
%U A004490 288807105787200,2021649740510400,6064949221531200,224403121196654400
%N A004490 Colossally abundant numbers: m for which there is a positive exponent epsilon such that sigma(m)/m^{1 + epsilon} >= sigma(k)/k^{1 + epsilon} for all k > 1, so that m attains the maximum value of sigma(m)/m^{1 + epsilon}.
%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.
%H A004490 Amiram Eldar, <a href="/A004490/b004490.txt">Table of n, a(n) for n = 1..382</a> (terms 1..150 from T. D. Noe)
%H A004490 Hirotaka Akatsuka, <a href="https://arxiv.org/abs/2411.19259">Maximal order for divisor functions and zeros of the Riemann zeta-function</a>, arXiv:2411.19259 [math.NT], 2024. See p. 4.
%H A004490 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 A004490 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 A004490 Kevin Broughan, <a href="https://doi.org/10.1017/9781108178228.008">A Variety of Abundant Numbers</a>, in Equivalents of the Riemann Hypothesis. Cambridge University Press, 2017, pp. 144-164.
%H A004490 Geoffrey Caveney, J.-L. Nicolas and J. Sondow, <a href="http://arxiv.org/abs/1112.6010">On SA, CA, and GA numbers</a>, arXiv:1112.6010 [math.NT], 2011-2012; Ramanujan J., 29 (2012), 359-384.
%H A004490 Geoffrey 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 A004490 J. C. Lagarias, <a href="https://arxiv.org/abs/math/0008177">An elementary problem equivalent to the Riemann hypothesis</a>, arXiv:math/0008177 [math.NT], 2000-2001; Am. Math. Monthly 109 (#6, 2002), 534-543.
%H A004490 S. Nazardonyavi and 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 A004490 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 A004490 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 A004490 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 A004490 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ColossallyAbundantNumber.html">Colossally Abundant Number</a>.
%F A004490 a(n) = Product_{k=1..n} A073751(k). - _Jeppe Stig Nielsen_, Nov 28 2021
%Y A004490 A subsequence of A004394 (superabundant numbers).
%Y A004490 Cf. A000203, A002201, A073751.
%Y A004490 Cf. A002093 (highly abundant numbers), A002182, A005101 (abundant numbers), A006038, A189228 (superabundant numbers that are not colossally abundant).
%K A004490 nonn
%O A004490 1,1
%A A004490 _N. J. A. Sloane_, Jan 22 2001