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

2,1

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A293557 A123587 A018840 * A160789 A266111 A131291

Adjacent sequences: A058206 A058207 A058208 * A058210 A058211 A058212

Keyword

sign,nice,easy

Author

N. J. A. Sloane, Nov 30 2000

Extensions

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

Status

approved