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).

2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800, 13967553600, 321253732800, 2248776129600, 65214507758400, 195643523275200, 6064949221531200, 12129898443062400, 448806242393308800, 18401055938125660800, 791245405339403414400

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• changed

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

Examples

			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
		

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. 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).

Links

• Iain Fox, Table of n, a(n) for n = 1..400 (first 150 terms from T. D. Noe)
• Hirotaka Akatsuka, Maximal order for divisor functions and zeros of the Riemann zeta-function, arXiv:2411.19259 [math.NT], 2024. See p. 4.
• 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

Crossrefs

Cf. A000705, A004490, A000005.

Cf. A002182, A072938, A106037, A094348, A003418, A002110, A263572.

Programs

Mathematica

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

Extensions

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