A058209 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).

-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

Offset: 2

Author

N. J. A. Sloane, Nov 30 2000

Keywords

• sign
• nice
• easy

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.

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.

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.
• G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384.

Crossrefs

Cf. A000203, A001620, A057641, A057642, A058210.

Programs

Maple

with(numtheory); Digits := 100; g := evalf(gamma); [seq( floor(exp(g)*n*log(log(n)))-sigma[1](n), n=2..80)];

Mathematica

a[n_] := Floor[Exp[EulerGamma] n*Log[Log[n]]] - DivisorSigma[1, n]; Array[a,100,2] (* Jean-François Alcover, May 04 2011 *)

PARI

a(n)=floor( exp(Euler)*n*log(log(n)) - sigma(n)) \\ Charles R Greathouse IV, Feb 08 2017

Extensions

Statement of Robin's theorem corrected by Jonathan Sondow, May 30 2011