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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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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.
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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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(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).
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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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STATUS
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approved
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