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}.

2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800, 160626866400, 321253732800, 9316358251200, 288807105787200, 2021649740510400, 6064949221531200, 224403121196654400

Offset: 1

Author

N. J. A. Sloane, Jan 22 2001

Keywords

• nonn

References

• 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.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..382 (terms 1..150 from T. D. Noe)
• Hirotaka Akatsuka, Maximal order for divisor functions and zeros of the Riemann zeta-function, arXiv:2411.19259 [math.NT], 2024. See p. 4.
• L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata
• Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 16 (2006), p. 251-256.
• Kevin Broughan, A Variety of Abundant Numbers, in Equivalents of the Riemann Hypothesis. Cambridge University Press, 2017, pp. 144-164.
• Geoffrey Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv:1112.6010 [math.NT], 2011-2012; Ramanujan J., 29 (2012), 359-384.
• Geoffrey Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv preprint arXiv:1112.6010 [math.NT], 2011. - From _N. J. A. Sloane_, Apr 14 2012
• J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, arXiv:math/0008177 [math.NT], 2000-2001; Am. Math. Monthly 109 (#6, 2002), 534-543.
• S. Nazardonyavi and S. Yakubovich, Extremely Abundant Numbers and the Riemann Hypothesis, Journal of Integer Sequences, 17 (2014), Article 14.2.8.
• S. Ramanujan, Highly composite numbers, Annotated and with a foreword by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119-153.
• T. Schwabhäuser, Preventing Exceptions to Robin's Inequality, arXiv preprint arXiv:1308.3678 [math.NT], 2013.
• M. Waldschmidt, From highly composite numbers to transcendental number theory, 2013.
• Eric Weisstein's World of Mathematics, Colossally Abundant Number.

Crossrefs

A subsequence of A004394 (superabundant numbers).

Cf. A000203, A002201, A073751.

Cf. A002093 (highly abundant numbers), A002182, A005101 (abundant numbers), A006038, A189228 (superabundant numbers that are not colossally abundant).

Formula

a(n) = Product_{k=1..n} A073751(k). - Jeppe Stig Nielsen, Nov 28 2021