1,2

Where record values of d(n) occur: d(n) > d(k) for all k < n.

RECORDS transform of A000005.

Flammenkamp's page has also a copy of the missing Siano paper.

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

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

Contribution from Bill R McEachen (bmceache(AT)centralsan.org), Feb 09 2009: (Start)

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....

An example is a(37)=665280, which is P(12,6)=12!/(12-6)! (End)

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]

A large number of highly composite numbers have 9 as their digit root. [From Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Jun 07 2009]

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]

CRC Press Standard Mathematical Tables 28th Ed p.61 [From Bill R McEachen (bmceache(AT)centralsan.org), Feb 09 2009]

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 180, p. 56, Ellipses, Paris 2008.

L. E. Dickson, History of Theory of Numbers, I, p. 323.

R. Honsberger, An introduction to Ramanujan's Highly Composite Numbers, Chap. 14 pp. 193-200 Mathematical Gems III, DME no. 9 MAA 1985

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. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.

S. Ratering, An interesting subset of the highly composite numbers, Math. Mag., 64 (1991), 343-346.

G. Robin, Methodes d'optimisation pour un probleme de theorie des nombres, RAIRO Informatique Theorique, 17, 1983, 239-247.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 128.

T. D. Noe, Table of n, a(n) for n = 1..1000

A. Flammenkamp, Highly composite numbers

A. Flammenkamp, List of the first 1200 highly composite numbers

J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543.

W. Lauritzen, Versatile Numbers -Versatile Economics

R. J. Mathar, Maple program to convert the Flammenkamp file to an OEIS b-file

R. J. Mathar, Output of above Maple program [Uncompresses to 9.1 MB]

Graeme McRae, Highly Composite Numbers

J.-L. Nicolas, Ordre maximal d'un element du groupe S_n de permutations et 'highly composite numbers' (Text in French)

J.-L. Nicolas and G. Robin, Highly Composite Numbers by Srinivasa Ramanujan, The Ramanujan Journal, Vol. 1(2), pp. 119-153, Kluwer Academics Pub.

K. O'Bryant, PlanetMath.org, Highly composite number

S. Ramanujan, Highly Composite Numbers

D. B. Siano and J. D. Siano, An Algorithm for Generating Highly Composite Numbers (pdf)

N. J. A. Sloane, Transforms

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Wikipedia, Highly composite number

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

a=0; Do[b=DivisorSigma[0, n]; If[b>a, a=b; Print[n]], {n, 1, 10^7}]

Cf. A000005, A002110, A002183, A002473, A004394, A106037.

Cf. A108602, A112778, A112779, A112780, A112781.

Cf. A006218, A126098.

Cf. A002201, A072938, A094348, A003418.

A161184 [From Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Jun 07 2009]

Sequence in context: A141420 A141551 A094348 this_sequence A077006 A004394 A166981

Adjacent sequences: A002179 A002180 A002181 this_sequence A002183 A002184 A002185

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).

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.

Additional references from Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 24 2001