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A067698
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Positive integers such that sigma(n) >= exp(gamma) * n * log(log(n)).
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16
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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
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OFFSET
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1,1
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COMMENTS
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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).
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REFERENCES
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Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.
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LINKS
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Table of n, a(n) for n=1..27.
G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), #A33.
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv:1112.6010 [math.NT], 2011-2012; Ramanujan J., 29 (2012), 359-384.
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, arXiv:math/0008177 [math.NT], 2000-2001; Am. Math. Monthly 109 (#6, 2002), 534-543.
Eric Weisstein's World of Mathematics, Gronwall's Theorem
Eric Weisstein's World of Mathematics, Robin's Theorem
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EXAMPLE
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9 is in the sequence since sigma(9) = 13 > 12.6184... = exp(gamma) * 9 * log(log(9)).
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A057641 (based on Lagarias' extension of Robin's result).
Cf. A091901, A189686, A004394, A196229.
Sequence in context: A093863 A337800 A091902 * A110495 A052347 A193299
Adjacent sequences: A067695 A067696 A067697 * A067699 A067700 A067701
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KEYWORD
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nonn,nice
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AUTHOR
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Ulrich Schimke (ulrschimke(AT)aol.com)
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EXTENSIONS
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Edited by N. J. A. Sloane at the suggestion of Max Alekseyev, Jul 17 2007
New name from Jud McCranie, Aug 14 2017
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STATUS
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approved
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