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A058209
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[ exp(gamma) n log log n ] - sigma(n), where gamma is Euler's constant (A001620) and sigma(n) is sum of divisors of n (A000203).
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6
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-5, -4, -5, -2, -6, 0, -5, -1, -4, 5, -9, 7, 0, 2, -2, 13, -5, 16, -3, 9, 8, 22, -11, 21, 12, 17, 4, 32, -7, 36, 7, 25, 22, 31, -10, 46, 27, 34, 2, 53, 2, 57, 20, 29, 37, 64, -9, 61, 28, 52, 29, 76, 13, 63, 18, 61, 54, 87, -18, 91, 60, 55, 35, 81, 24, 103, 48, 81, 36, 111, -9, 115
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,1
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COMMENTS
| Theorem (G. Robin): exp(gamma) n log log n - sigma(n) is positive for all n >= 5041 if and only if the Riemann Hypothesis is true.
Note that a(n) <= exp(gamma) n log log n - sigma(n) < a(n) + 1.
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REFERENCES
| D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.2.2.b.
G. 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
| T. D. Noe, Table of n, a(n) for n=2..10000
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.
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MAPLE
| with(numtheory); Digits := 100; g := evalf(gamma); [seq( floor(exp(g)*n*log(log(n)))-sigma[1](n), n=2..80)];
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MATHEMATICA
| a[n_] := Floor[Exp[EulerGamma] n*Log[Log[n]]] - DivisorSigma[1, n]; Array[a, 100, 2] (* From Jean-François Alcover, May 4 2011 *)
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CROSSREFS
| Cf. A000203, A001620, A057641, A057642, A058210.
Sequence in context: A204372 A123587 A018840 * A160789 A131291 A131369
Adjacent sequences: A058206 A058207 A058208 * A058210 A058211 A058212
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KEYWORD
| sign,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Nov 30 2000
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EXTENSIONS
| Statement of Robin's theorem corrected by Jonathan Sondow (jsondow(AT)alumni.princeton.edu), May 30 2011
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