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A067698 Positive integers such that sigma(n) >= exp(gamma) * n * log(log(n)). 16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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).
REFERENCES
Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.
LINKS
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
EXAMPLE
9 is in the sequence since sigma(9) = 13 > 12.6184... = exp(gamma) * 9 * log(log(9)).
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 *)
PROG
(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
CROSSREFS
Cf. A057641 (based on Lagarias' extension of Robin's result).
Sequence in context: A093863 A337800 A091902 * A110495 A367501 A367463
KEYWORD
nonn,nice
AUTHOR
Ulrich Schimke (ulrschimke(AT)aol.com)
EXTENSIONS
Edited by N. J. A. Sloane at the suggestion of Max Alekseyev, Jul 17 2007
New name from Jud McCranie, Aug 14 2017
STATUS
approved

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Last modified March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)