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A067698
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Numbers with relatively many and large divisors (see comments).
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9
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| 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
| Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.
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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.
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, 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
| 9 is in the sequence since sigma(9) = 13 > 12.6184... = exp(gamma) * 9 * log(log(9)).
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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:
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MATHEMATICA
| fQ[n_] := DivisorSigma[1, n] > n*Exp@ EulerGamma*Log@ Log@n; lst = {}; Do[ If[ fQ@n, Print@n; AppandTo[lst, n]], {n, 10^9}] (* Robert G. Wilson v *)
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CROSSREFS
| Cf. A057641 based on Lagarias' extension of Robin's result.
Cf. A091901, A189686, A004394, A196229.
Sequence in context: A173786 A093863 A091902 * A110495 A052347 A193299
Adjacent sequences: A067695 A067696 A067697 * A067699 A067700 A067701
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KEYWORD
| nonn,nice
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AUTHOR
| Ulrich Schimke (ulrschimke(AT)aol.com)
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion of Max Alekseyev, Jul 17 2007
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