A263572 -id:A263572 - cp's OEIS frontend

A328549 1, together with the numbers that are simultaneously superior highly composite (A002201), colossally abundant (A004490), deeply composite (A095848), and miserable average divisor numbers (A263572).

1, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440

Offset: 1

N. J. A. Sloane, Oct 20 2019

Presumably there are no further terms.
From Hal M. Switkay, Nov 04 2019: (Start)
1. a(n+1) is the product of the first n terms of A328852.
2. This sequence is most rapidly constructed as the intersection of A095849 and A224078. It is designed to list all potential solutions to a question. Let n be a natural number, k real <= 0, e real > 0. Let P(n,k,e) state: on the domain of natural numbers, sigma_k(x)/x^e reaches a maximum at x = n. This implies Q(n,k): sigma_k(n) > sigma_k(m) for m < n a natural number. We ask: for which natural numbers n is it true for all real k <= 0 that there is a real e > 0 such that P(n,k,e)?
If any such n exist, they must belong to the present sequence. A095849 consists of all natural numbers n such that for all real k <= 0, Q(n,k) holds. A224078 consists of all natural numbers n such that for some real e0 and e1 both > 0, P(n,0,e0) and P(n,-1,e1) hold. It would be interesting to see the list of n for which there is an e2 > 0 such that P(n,-2,e2) holds.
Conjecture: the solutions to this problem, if any, form an initial sequence of the present sequence. (End)
Every term of this sequence is also in A065385: a record for the cototient function. - Hal M. Switkay, Feb 27 2021
Every term of this sequence, except the first, is also in A210594: factor-dense numbers. - Hal M. Switkay, Mar 29 2021

Hal M. Switkay, Email to N. J. A. Sloane, Oct 20 2019

1 together with the intersection of A002201, A004490, A095848, A263572.

Cf. A095849, A224078, A328852, A210594.

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).

Original entry on oeis.org

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

N. J. A. Sloane

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

			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

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)
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

Cf. A000705, A004490, A000005.

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

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

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

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

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A328549 1, together with the numbers that are simultaneously superior highly composite (A002201), colossally abundant (A004490), deeply composite (A095848), and miserable average divisor numbers (A263572).

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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).

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