login
This site is supported by donations to The OEIS Foundation.

 

Logo

Many excellent designs for a new banner were submitted. We will use the best of them in rotation.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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).
(Formerly M1591 N0620)
18
2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800, 13967553600, 321253732800, 2248776129600, 65214507758400, 195643523275200, 6064949221531200 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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, Sep 29 2004

REFERENCES

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.

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).

LINKS

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

EXAMPLE

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.

Ant King asks if the intervals in the previous line can be extended to include the left endpoints. May 02 2005

CROSSREFS

Cf. A000705, A004490, A000005.

Cf. A002182, A072938, A106037, A094348, A003418, A002110.

Sequence in context: A065887 A072181 A126915 * A004490 A224078 A135060

Adjacent sequences:  A002198 A002199 A002200 * A002202 A002203 A002204

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Better definition from T. D. Noe, Nov 05 2002

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified April 24 12:03 EDT 2014. Contains 240983 sequences.