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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}. 54
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
Amiram Eldar, Table of n, a(n) for n = 1..382 (terms 1..150 from T. D. Noe)
L. Alaoglu and P. Erdős, 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 [math.NT], 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 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.
Eric Weisstein's World of Mathematics, Colossally Abundant Number.
FORMULA
a(n) = Product_{k=1..n} A073751(k). - Jeppe Stig Nielsen, Nov 28 2021
CROSSREFS
A subsequence of A004394 (superabundant numbers).
Cf. A002093 (highly abundant numbers), A002182, A005101 (abundant numbers), A006038, A189228 (superabundant numbers that are not colossally abundant).
Sequence in context: A328549 A002201 A263572 * A224078 A309811 A309233
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 22 2001
STATUS
approved

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Last modified May 19 14:45 EDT 2024. Contains 372698 sequences. (Running on oeis4.)