Search: id:A002201
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%I A002201 M1591 N0620
%S A002201 2,6,12,60,120,360,2520,5040,55440,720720,1441440,4324320,
%T A002201 21621600,367567200,6983776800,13967553600,321253732800,
%U A002201 2248776129600,65214507758400,195643523275200,6064949221531200
%N A002201 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).
%C A002201 For fixed e > 0, d(n)/n^e is bounded and reaches its maximum at one or 
               more points.
%C A002201 This is an infinite subset of A002182.
%C A002201 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
%D A002201 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002201 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002201 J. L. Nicolas, On highly composite numbers, pp. 215-244 in Ramanujan 
               Revisited, Editors G. E. Andrews et al., Academic Press 1988.
%D A002201 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.
%D A002201 S. Ramanujan, Highly composite numbers, Annotated and with a foreword 
               by J.-L. Nicholas and G. Robin, Ramanujan J., 1 (1997), 119-153.
%D A002201 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.
%D A002201 S. Ratering, An interesting subset of the highly composite numbers, Math. 
               Mag., 64 (1991), 343-346.
%H A002201 T. D. Noe, <a href="b002201.txt">Table of n, a(n) for n = 1..150</a>
%H A002201 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/
               Cpaper15/page35.htm">IV: Superior Highly Composite Numbers</a>
%H A002201 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SuperiorHighlyCompositeNumber.html">Superior Highly Composite Number</
               a>
%H A002201 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ColossallyAbundantNumber.html">Colossally Abundant Number</a>
%H A002201 Wikipedia, <a href="http://en.wikipedia.org/wiki/Superior_highly_composite_number">
               Superior highly composite number</a>
%e A002201 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.
%e A002201 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
%Y A002201 Cf. A000705, A004490, A000005.
%Y A002201 Cf. A002182, A072938, A106037, A094348, A003418, A002110.
%Y A002201 Sequence in context: A065887 A072181 A126915 this_sequence A004490 A135060 
               A072486
%Y A002201 Adjacent sequences: A002198 A002199 A002200 this_sequence A002202 A002203 
               A002204
%K A002201 nonn,nice
%O A002201 1,1
%A A002201 N. J. A. Sloane (njas(AT)research.att.com).
%E A002201 Better definition from T. D. Noe (noe(AT)sspectra.com), Nov 05 2002

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