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Search: id:A002201
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| A002201 |
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Superior highly composite numbers: positive integers n for which there is an e>0 such that d(n)/n^e >= d(k)/k^e for all k>1, where the function d(n) counts the divisors of n (A000005).
(Formerly M1591 N0620)
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+0
9
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| 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800, 13967553600, 321253732800, 2248776129600, 65214507758400, 195643523275200, 6064949221531200
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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For fixed e > 0, d(n)/n^e is bounded and reaches its maximum at one or more points.
This is an infinite subset of A002182.
The first 15 numbers in this sequence agree with those in A004490 (colossally abundant numbers). - David Terr (David_C_Terr(AT)raytheon.com), Sep 29 2004
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REFERENCES
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J. L. Nicolas, On highly composite numbers, pp. 215-244 in Ramanujan Revisited, Editors G. E. Andrews et al., Academic Press 1988.
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.
S. Ramanujan, Highly Composite Numbers: Section IV, in 1) Collected Papers of Srinivasa Ramanujan, pp. 111-8, Ed. G. H. Hardy et al., AMS Chelsea 2000. 2) Ramanujan's Papers, pp. 143-150, Ed. B. J. Venkatachala et al., Prism Books Bangalore 2000.
S. Ratering, An interesting subset of the highly composite numbers, Math. Mag., 64 (1991), 343-346.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..150
S. Ramanujan, IV: Superior Highly Composite Numbers
Eric Weisstein's World of Mathematics, Superior Highly Composite Number
Eric Weisstein's World of Mathematics, Colossally Abundant Number
Wikipedia, Superior highly composite number
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EXAMPLE
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For n=2, 6, and 12 we may take e in the intervals (log(2)/log(3), 1], (log(3/2)/log(2), log(2)/log(3)], and (log(2)/log(5), log(3/2)/log(2)], respectively.
A correspondent ("mathstutoring(AT)ntlworld.com") asks if the intervals in the previous line can be extended to include the left endpoints. May 02 2005
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CROSSREFS
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Cf. A000705, A004490, A000005.
Sequence in context: A065887 A072181 A126915 this_sequence A004490 A135060 A072486
Adjacent sequences: A002198 A002199 A002200 this_sequence A002202 A002203 A002204
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KEYWORD
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nonn,nice
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AUTHOR
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njas
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EXTENSIONS
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Better definition from T. D. Noe (noe(AT)sspectra.com), Nov 05 2002
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