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Search: id:A004490
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| A004490 |
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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}. |
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13
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| 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800, 160626866400, 321253732800, 9316358251200, 288807105787200, 2021649740510400, 6064949221531200, 224403121196654400
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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All superabundant, colossally abundant, highly composite and superior highly composite numbers are Niven/Harshad numbers. - Rob Hoogers (chimera(AT)chimera.fol.nl), Jun 26 2004
The previous comment is erroneous. The first superabundant number that is not a Harshad number is A004394(105) = 149602080797769600. The first highly composite number that is not a Harshad number is A002182(61) = 245044800. For all exceptions I found, the sum of digits is a power of 3. Although the first 60000 terms of the colossally abundant numbers and the superior highly composite numbers are Harshad numbers, I am not aware of a proof that all terms are Harshad numbers. There may be large counterexamples. [From T. D. Noe (noe(AT)sspectra.com), Oct 27 2009]
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REFERENCES
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L. Alaoglu and P. Erdos, On Highly Composite and Similar Numbers, Trans. Amer. Math. Soc. 56 (1944), 448-469.
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.
S. Ramanujan, Highly composite numbers, Annotated and with a foreword by J.-L. Nicholas and G. Robin, Ramanujan J., 1 (1997), 119-153.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..150
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543.
Eric Weisstein's World of Mathematics, Colossally Abundant Number
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CROSSREFS
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A subset of A004394. Cf. A002201.
Cf. A073751.
Cf. abundant numbers = A002093, A002182, A005101, A006038, A004394; colossally abundant numbers = A004490, highly abundant numbers = A002093, superabundant numbers = A004394.
Sequence in context: A072181 A126915 A002201 this_sequence A135060 A072486 A096123
Adjacent sequences: A004487 A004488 A004489 this_sequence A004491 A004492 A004493
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Jan 22 2001
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