%I A002182 M1025 N0385
%S A002182 1,2,4,6,12,24,36,48,60,120,180,240,360,720,840,1260,1680,2520,5040,
%T A002182 7560,10080,15120,20160,25200,27720,45360,50400,55440,83160,110880,166320,
%U A002182 221760,277200,332640,498960,554400,665280,720720,1081080,1441440,2162160
%N A002182 Highly composite numbers, definition (1): where d(n), the number of divisors 
               of n (A000005), increases to a record.
%C A002182 Where record values of d(n) occur: d(n) > d(k) for all k < n.
%C A002182 RECORDS transform of A000005.
%C A002182 Flammenkamp's page has also a copy of the missing Siano paper.
%C A002182 Highly composite numbers are the product of primorials, A002110. See 
               A112779 for the number of primorial terms in the product of a highly 
               composite number. - Jud McCranie (j.mccranie(AT)comcast.net), Jun 
               12 2005
%C A002182 Sigma and tau for highly composite numbers through the 146th entry conform 
               to a power fit as follows: ln(sigma)=A*ln(tau)^B where (A,B) =~ (1.45,
               1.38). - Bill McEachen (bmceache(AT)centralsan.dst.ca.us), May 24 
               2006
%C A002182 Contribution from Bill R McEachen (bmceache(AT)centralsan.org), Feb 09 
               2009: (Start)
%C A002182 a(n) often corresponds to P(n,m) = number of permutations of n things 
               taken m at a time. Specifically, if start=1, pointers 1-6,9,10,13-15,
               17-19,22,23,28,34,37,43,52....
%C A002182 An example is a(37)=665280, which is P(12,6)=12!/(12-6)! (End)
%C A002182 Concerning the previous comment, if m=1, then P(n,m) can represent any 
               number. So let's assume m>1. Searching the first 1000 terms, the 
               only indices of terms of the form P(n,m) are 4, 5, 6, 9, 10, 12, 
               13, 14, 15, 16, 17, 18, 19, 22, 23, 27, 28, 31, 34, 37, 41, 43, 44, 
               47, 50, 52, and 54. Note that a(44) = 4324320 = P(2079,2). See A163264. 
               [From T. D. Noe (noe(AT)sspectra.com), Jun 10 2009]
%C A002182 A large number of highly composite numbers have 9 as their digit root. 
               [From Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Jun 07 2009]
%C A002182 Because 9 divides all highly composite numbers greater than 1680, those 
               numbers have digital root 9. [From T. D. Noe (noe(AT)sspectra.com), 
               Jul 24 2009]
%D A002182 CRC Press Standard Mathematical Tables 28th Ed p.61 [From Bill R McEachen 
               (bmceache(AT)centralsan.org), Feb 09 2009]
%D A002182 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 180, p. 56, Ellipses, 
               Paris 2008.
%D A002182 L. E. Dickson, History of Theory of Numbers, I, p. 323.
%D A002182 R. Honsberger, An introduction to Ramanujan's Highly Composite Numbers, 
               Chap. 14 pp. 193-200 Mathematical Gems III, DME no. 9 MAA 1985
%D A002182 J. L. Nicolas, On highly composite numbers, pp. 215-244 in Ramanujan 
               Revisited, Editors G. E. Andrews et al., Academic Press 1988
%D A002182 S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 
               347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 
               1927; Chelsea, NY, 1962.
%D A002182 S. Ratering, An interesting subset of the highly composite numbers, Math. 
               Mag., 64 (1991), 343-346.
%D A002182 G. Robin, Methodes d'optimisation pour un probleme de theorie des nombres, 
               RAIRO Informatique Theorique, 17, 1983, 239-247.
%D A002182 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002182 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002182 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. 
               Penguin Books, NY, 1986, 128.
%H A002182 T. D. Noe, <a href="b002182.txt">Table of n, a(n) for n = 1..1000</a>
%H A002182 A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.html">
               Highly composite numbers</a>
%H A002182 A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.txt">
               List of the first 1200 highly composite numbers</a>
%H A002182 J. C. Lagarias, <a href="http://arXiv.org/abs/math.NT/0008177">An elementary 
               problem equivalent to the Riemann hypothesis</a>, Am. Math. Monthly 
               109 (#6, 2002), 534-543.
%H A002182 W. Lauritzen, <a href="http://www.earth360.com/math-versatile.html">Versatile 
               Numbers -Versatile Economics</a>
%H A002182 R. J. Mathar, <a href="a002182.mp">Maple program to convert the Flammenkamp 
               file to an OEIS b-file</a>
%H A002182 R. J. Mathar, <a href="a002182.txt.gz">Output of above Maple program</
               a> [Uncompresses to 9.1 MB]
%H A002182 Graeme McRae, <a href="http://2000clicks.com/MathHelp/NumberFactorsHighlyComposite.htm">
               Highly Composite Numbers</a>
%H A002182 J.-L. Nicolas, <a href="http://archive.numdam.org/article/BSMF_1969__97__129_0.pdf">
               Ordre maximal d'un element du groupe S_n de permutations et 'highly 
               composite numbers' (Text in French)</a>
%H A002182 J.-L. Nicolas and G. Robin, <a href="http://www.wkap.nl/oasis.htm/129255">
               Highly Composite Numbers by Srinivasa Ramanujan</a>, The Ramanujan 
               Journal, Vol. 1(2), pp. 119-153, Kluwer Academics Pub.
%H A002182 K. O'Bryant, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/
               HighlyCompositeNumber.html">Highly composite number</a>
%H A002182 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/
               Cpaper15/page1.htm">Highly Composite Numbers</a>
%H A002182 D. B. Siano and J. D. Siano, An Algorithm for Generating Highly Composite 
               Numbers (<a href="http://www.eclipse.net/~dimona/julianmanuscript3.pdf">
               pdf</a>)
%H A002182 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%H A002182 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HighlyCompositeNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A002182 Wikipedia, <a href="http://www.wikipedia.org/wiki/Highly_composite_number">
               Highly composite number</a>
%F A002182 Also, for n >=2, smallest values of p for which a006218(p)-A006318(p-1)=A002183(n) 
               - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 23 2007
%t A002182 a=0; Do[b=DivisorSigma[0, n]; If[b>a, a=b; Print[n]], {n, 1, 10^7}]
%Y A002182 Cf. A000005, A002110, A002183, A002473, A004394, A106037.
%Y A002182 Cf. A108602, A112778, A112779, A112780, A112781.
%Y A002182 Cf. A006218, A126098.
%Y A002182 Cf. A002201, A072938, A094348, A003418.
%Y A002182 A161184 [From Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Jun 07 2009]
%Y A002182 Sequence in context: A141420 A141551 A094348 this_sequence A077006 A004394 
               A166981
%Y A002182 Adjacent sequences: A002179 A002180 A002181 this_sequence A002183 A002184 
               A002185
%K A002182 nonn,nice,easy
%O A002182 1,2
%A A002182 N. J. A. Sloane (njas(AT)research.att.com).
%E A002182 Jun 19 1996: Changed beginning to start at 1. Jul 10 1996: Matthew Conroy 
               points out that these are different from the super-abundant numbers 
               - see A004394. Last 8 terms sent by J. Lowell, jhbubby(AT)avana.net; 
               checked by Jud McCranie (j.mccranie(AT)comcast.net). Description 
               corrected by Gerard Schildberger and N. J. A. Sloane (njas(AT)research.att.com), 
               Apr 04 2001.
%E A002182 Additional references from Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 
               24 2001