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%I #28 Oct 18 2022 17:59:39
%S -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,
%T 17,4,32,-7,36,7,25,22,31,-10,46,27,34,2,53,2,57,20,29,37,64,-9,61,28,
%U 52,29,76,13,63,18,61,54,87,-18,91,60,55,35,81,24,103,48,81,36,111,-9,115
%N a(n) = floor( 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).
%C 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.
%C Note that a(n) <= exp(gamma) n log log n - sigma(n) < a(n) + 1.
%D D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.2.2.b.
%D G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.
%H T. D. Noe, <a href="/A058209/b058209.txt">Table of n, a(n) for n = 2..10000</a>
%H G. Caveney, J.-L. Nicolas, and J. Sondow, <a href="http://www.integers-ejcnt.org/l33/l33.pdf">Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis</a>, Integers 11 (2011), #A33.
%H G. Caveney, J.-L. Nicolas and J. Sondow, <a href="http://arxiv.org/abs/1112.6010">On SA, CA, and GA numbers</a>, Ramanujan J., 29 (2012), 359-384.
%p with(numtheory); Digits := 100; g := evalf(gamma); [seq( floor(exp(g)*n*log(log(n)))-sigma[1](n), n=2..80)];
%t a[n_] := Floor[Exp[EulerGamma] n*Log[Log[n]]] - DivisorSigma[1, n]; Array[a,100,2] (* _Jean-François Alcover_, May 04 2011 *)
%o (PARI) a(n)=floor( exp(Euler)*n*log(log(n)) - sigma(n)) \\ _Charles R Greathouse IV_, Feb 08 2017
%Y Cf. A000203, A001620, A057641, A057642, A058210.
%K sign,nice,easy
%O 2,1
%A _N. J. A. Sloane_, Nov 30 2000
%E Statement of Robin's theorem corrected by _Jonathan Sondow_, May 30 2011
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