1,1

n is in the sequence iff sigma(n) >= exp(gamma) * n * log(log(n)), where gamma is the 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 (sigma(n)/(n*log(log(n)))) = exp(gamma)

Guy Robin: Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann, J. Math. Pures Appl. 63 (1984), 187-213

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, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Robin's Theorem

9 is in the sequence since sigma(9) = 13 > 12.6184... = exp(gamma) * 9* log(log(9))

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:

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 *)

Cf. A057641 based on Lagarias' extension of Robin's result.

Cf. A091901.

Sequence in context: A048645 A093863 A091902 this_sequence A110495 A052347 A022773

Adjacent sequences: A067695 A067696 A067697 this_sequence A067699 A067700 A067701

nonn,nice

Ulrich Schimke (ulrschimke(AT)aol.com)

Edited by njas at the suggestion of Max Alekseyev, Jul 17 2007