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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}. 25
2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800, 160626866400, 321253732800, 9316358251200, 288807105787200, 2021649740510400, 6064949221531200, 224403121196654400 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

T. D. Noe, Table of n, a(n) for n = 1..150

L. Alaoglu and P. Erdos, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata

G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384.

Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 16 (2006), p. 251-256.

G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv preprint arXiv:1112.6010, 2011. - From N. J. A. Sloane, Apr 14 2012

J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543.

S. Nazardonyavi, 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. Nicholas and G. Robin, Ramanujan J., 1 (1997), 119-153.

T. Schwabhäuser, Preventing Exceptions to Robin's Inequality, arXiv preprint arXiv:1308.3678, 2013

M. Waldschmidt, From highly composite numbers to transcendental number theory, 2013.

Eric Weisstein's World of Mathematics, Colossally Abundant Number

CROSSREFS

A subset of A004394. Cf. A002201.

Cf. A073751.

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

Sequence in context: A072181 A126915 A002201 * A224078 A135060 A185021

Adjacent sequences:  A004487 A004488 A004489 * A004491 A004492 A004493

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jan 22 2001

STATUS

approved

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Last modified November 27 21:59 EST 2014. Contains 250286 sequences.