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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).


Iain Fox, Table of n, a(n) for n = 1..400 (first 150 terms from T. D. Noe)

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


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


(PARI) lista(nn)=my(p=primes(primepi(2^log(nn)))); setminus(Set(vector(nn, i, prod(n=1, primepi(2^log(i)), p[n]^floor(1/(p[n]^(1/log(i))-1))))), [1]) \\ Iain Fox, Aug 23 2020


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




N. J. A. Sloane


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



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Last modified September 18 07:39 EDT 2020. Contains 337166 sequences. (Running on oeis4.)