for every integer (Robin 1984; Havil 2003, p. 207), where is the divisor
function giving the sum of the positivedivisors
of and is the Euler-Mascheroni
constant. For integers, the inequality holds for 7, 11, 13, 14, 15, 17, 19,
... (OEIS A091901), and fails at 2, 3, 4, 5,
6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 120, 180, 240, 360, 720,
840, 2520, and 5040 (OEIS A067698).
Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.Nicolas,
J.-L. "Petites valeurs de la fonction d'Euler." J. Number Th.17,
375-388, 1983.Robin, G. "Grandes valeurs de la fonction somme des
diviseurs et hypothèse de Riemann." J. Math. Pures Appl.63,
187-213, 1984.Schoenfeld, L. "Sharper Bounds for the Chebyshev
Functions
and .
II." Math. Comput.30, 337-360, 1976.Sloane, N. J. A.
Sequences A067698 and A091901
in "The On-Line Encyclopedia of Integer Sequences."
(Robin 1984; Havil 2003, p. 207), where 
is the divisor
function giving the sum of the positive divisors
of 
and 
is the Euler-Mascheroni
constant. For integers 
, the inequality holds for 7, 11, 13, 14, 15, 17, 19,
... (OEIS A091901), and fails at 2, 3, 4, 5,
6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 120, 180, 240, 360, 720,
840, 2520, and 5040 (OEIS A067698).
and 
.
II." Math. Comput. 30, 337-360, 1976.Sloane, N. J. A.
Sequences