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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)
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)



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


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. Nicolas 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.

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).


T. D. Noe, Table of n, a(n) for n = 1..150

S. Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society, 2, XIV, 1915, 347 - 409.

S. Ramanujan, IV: Superior Highly Composite Numbers

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

Eric Weisstein's World of Mathematics, Superior Highly Composite Number

Eric Weisstein's World of Mathematics, Colossally Abundant Number

Wikipedia, Superior highly composite number


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.

Can the intervals in the previous line can be extended to include the left endpoints? - Ant King, May 02 2005


Cf. A000705, A004490, A000005.

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

Sequence in context: A065887 A072181 A126915 * A263572 A004490 A224078

Adjacent sequences:  A002198 A002199 A002200 * A002202 A002203 A002204




N. J. A. Sloane


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



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Last modified November 30 00:53 EST 2015. Contains 264663 sequences.