

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)


33



2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800, 13967553600, 321253732800, 2248776129600, 65214507758400, 195643523275200, 6064949221531200
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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. 215244 in Ramanujan Revisited, Editors G. E. Andrews et al., Academic Press 1988.
S. Ramanujan, Highly composite numbers, Proc. London Math. Soc., 14 (1915), 347407. Reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, pp. 78129. 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), 119153.
S. Ramanujan, Highly Composite Numbers: Section IV, in 1) Collected Papers of Srinivasa Ramanujan, pp. 1118, Ed. G. H. Hardy et al., AMS Chelsea 2000. 2) Ramanujan's Papers, pp. 143150, 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).


LINKS

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), 343346.
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.
Can the intervals in the previous line can be extended to include the left endpoints?  Ant King, May 02 2005


MATHEMATICA

Rest@ Union@ Array[Product[p^Floor[1/(p^(1/#)  1)], {p, Prime@ Range@ PrimePi[2^#]}] &[Log@ #] &, 160] (* Michael De Vlieger, Jul 09 2019 *)


CROSSREFS

Cf. A000705, A004490, A000005.
Cf. A002182, A072938, A106037, A094348, A003418, A002110.
Sequence in context: A322381 A265125 A328549 * A263572 A004490 A224078
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



