A067698 Positive integers such that sigma(n) >= exp(gamma) * n * log(log(n)).

2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 120, 180, 240, 360, 720, 840, 2520, 5040

Offset: 1

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Keywords

• nonn
• nice

Comments

Previous name was: Numbers with relatively many and large divisors.

n is in the sequence iff sigma(n) >= exp(gamma) * n * log(log(n)), where gamma = Euler-Mascheroni constant and sigma(n) = sum of divisors of n.

Robin has shown that 5040 is the last element in the sequence iff the Riemann hypothesis is true. Moreover the sequence is infinite if the Riemann hypothesis is false. Gronwall's theorem says that

lim sup_{n -> infinity} sigma(n)/(n*log(log(n))) = exp(gamma).

a(28) > 10^(10^13.11485), if the Riemann hypothesis is false (Morrill and Platt, 2021). Briggs (2006) found the lower bound 10^(10^10). - Amiram Eldar, Jul 23 2025

Examples

			9 is in the sequence since sigma(9) = 13 > 12.6184... = exp(gamma) * 9 * log(log(9)).
		

Links

• Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 15, No. 2 (2006), p. 251-256.
• Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), #A33.
• Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, On SA, CA, and GA numbers, The Ramanujan Journal, Vol. 29 (2012), pp. 359-384; arXiv preprint, arXiv:1112.6010 [math.NT], 2011-2012.
• Jeffrey C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, The American Mathematical Monthly, Vol. 109, No. 6 (2002), pp. 534-543; alternative copy; arXiv preprint, arXiv:math/0008177 [math.NT], 2000-2001.
• Thomas Morrill and David John Platt, Robin's inequality for 20-free integers, Integers, Vol. 21 (2021), #A28.
• Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.
• Eric Weisstein's World of Mathematics, Gronwall's Theorem.
• Eric Weisstein's World of Mathematics, Robin's Theorem.

Crossrefs

Cf. A057641 (based on Lagarias's extension of Robin's result).

Cf. A000203, A004394, A073004, A091901, A189686, A196229.

Programs

Maple

with (numtheory): expgam := exp(evalf(gamma)): for i from 2 to 6000 do: a := sigma (i): b := expgam*i*evalf(ln(ln(i))): if a >= b then print (i, a, b): fi: od:

Mathematica

fQ[n_] := DivisorSigma[1, n] > n*Exp@ EulerGamma*Log@ Log@n; lst = {}; Do[ If[ fQ[n], AppendTo[lst, n]], {n,2,10^4}]; lst (* Robert G. Wilson v, May 16 2003 *)
Select[Range[2,5050], Exp[EulerGamma] # Log[Log[#]]-DivisorSigma[1,#]<0 &] (* Ant King, Feb 28 2013 *)

PARI

is(n)=sigma(n) >= exp(Euler) * n * log(log(n)) \\ Charles R Greathouse IV, Feb 08 2017

Python

from sympy import divisor_sigma, EulerGamma, E, log
print([n for n in range(2, 5041) if divisor_sigma(n) >= (E**EulerGamma * n * log(log(n)))]) # Karl-Heinz Hofmann, Apr 22 2022

Extensions

Edited by N. J. A. Sloane at the suggestion of Max Alekseyev, Jul 17 2007

New name from Jud McCranie, Aug 14 2017