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A002182 Highly composite numbers, definition (1): where d(n), the number of divisors of n (A000005), increases to a record.
(Formerly M1025 N0385)
179
1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

A002183 is the RECORDS transform of A000005, i.e., lists the corresponding values d(n) for n in A002182.

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, 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, May 24 2006

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)!. - Bill McEachen, Feb 09 2009

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, Jun 10 2009]

A large number of highly composite numbers have 9 as their digit root. [From Parthasarathy Nambi, Jun 07 2009]

Because 9 divides all highly composite numbers greater than 1680, those numbers have digital root 9. [From T. D. Noe, Jul 24 2009]

See A181309 for highly composite numbers that are not highly abundant.

a(n) is also defined by the recurrence : a(1) = 1, a(n+1)/sigma(a(n+1)) < a(n) / sigma(a(n)) - [From Michel Lagneau, Jan 02 2012]. [This "definition" is wrong (the term a(20)=7560 does not satisfy this inequality) and incomplete: It does not determine a sequence uniquely, e.g., any subsequence would satisfy the same relation. The intended meaning is probably the definition of the (different) sequence A004394. - M. F. Hasler, Sep 13 2012]

Up to a(1000), the terms beyond a(5)=12 resp. beyond a(9)=60 are a multiples of these. Is this true for all subsequent terms? - M. F. Hasler, Sep 13 2012

Differs from the superabundant numbers from a(20)=7560 on, which is not in A004394. The latter is not a subsequence of A002182, as might appear from considering the displayed terms: The two sequences have only 449 terms in common, the largest of which is A002182(2567) = A004394(1023). See A166735 for superabundant numbers that are not highly composite, and A004394 for further information. - M. F. Hasler, Sep 13 2012

REFERENCES

CRC Press Standard Mathematical Tables 28th Ed p.61 [From Bill McEachen, 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. 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).

M. Waldschmidt, From highly composite numbers to transcendental number theory, 2013; http://www.math.jussieu.fr/~miw/articles/pdf/LegacyRamanujan2013Text.pdf

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

LINKS

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, Proceedings of the London Mathematical Society, 2, XIV, 1915, 347 - 409.

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, Highly Composite Number

Wikipedia, Highly composite number

FORMULA

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

MATHEMATICA

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

PROG

(PARI) print1(r=1); forstep(n=2, 1e5, 2, if(numdiv(n)>r, r=numdiv(n); print1(", "n))) \\ Charles R Greathouse IV, Jun 10 2011

CROSSREFS

Cf. A000005, A002110, A002183, A002473, A004394, A106037, A108602, A112778, A112779, A112780, A112781, A006218, A126098, A002201, A072938, A094348, A003418.

A161184 [From Parthasarathy Nambi, Jun 07 2009]

Sequence in context: A094348 A242298 A189686 * A077006 A166981 A004394

Adjacent sequences:  A002179 A002180 A002181 * A002183 A002184 A002185

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

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; checked by Jud McCranie. Description corrected by Gerard Schildberger and N. J. A. Sloane, Apr 04 2001.

Additional references from Lekraj Beedassy, Jul 24 2001

STATUS

approved

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Last modified October 25 13:27 EDT 2014. Contains 248536 sequences.