Arithmetic inequalities and the OEIS; was Mandl's ineq.

[N. J. A. Sloane]

If f(n) <= g(n) for n >= n_0 is an arithmetic inequality, like Mandl's,
then often [g(n)-f(n), n = 1,2,3,... ] or some variant of it is an
integer sequence that should be added to the OEIS.

The reason for this is so that someone else coming across
the same inequality would look up the sequence and, hopefully, discover
it in the OEIS and find a reference to a proof or disproof.

A random example from the Mitrinovic et al.
Handbook of Number Theory, Kluwer, Section I.3.1 (but they have "tau" instead of "sigma"):

Sum of divisors of n >= phi(n) - number of divisors of n, for n >= 2,

which gives the sequence

Sum of divisors of n - phi(n) - number of divisors of n, n >= 1,
0,-1,0,0,2,0,6,0,7,4,10,0,18,0,14,12,18,0,27,0,28,16,22,0,44,8,...
- see entry A046520 in the OEIS.

In the case of Mandl's inequality, I am adding this entry:

%I A124478
%S A124478 2,2,1,1,0,0,1,0,1,4,2,5,5,3,4,6,8,6,8,8,6,8,7,9,13,12,10,10,7,7,17,16,
%T A124478 17,14,19,17,18,19,18,19,20,17,22,19,18,16,23,29,28,25,24,25,21,26,27,28,
%U A124478 28,25,26,25,22,27,35,34,30,29,37,38,42,38,37,37,40,40,40,39,39,41,40,42
%V A124478 -2,-2,-1,-1,0,0,1,0,1,4,2,5,5,3,4,6,8,6,8,8,6,8,7,9,13,12,10,10,7,7,17,16,
%W A124478 17,14,19,17,18,19,18,19,20,17,22,19,18,16,23,29,28,25,24,25,21,26,27,28,
%X A124478 28,25,26,25,22,27,35,34,30,29,37,38,42,38,37,37,40,40,40,39,39,41,40,42
%N A124478 Let p(n) = prime(n); sequence gives p(n) - floor( (2/n)*(Sum_{i=1..n} p(i)) ).
%C A124478 Robert Mandl conjectured and Rosser and Schoenfeld proved that p(n)/2 > (Sum_{i=1..n} p(i))/n for n >= 9 (implying that a(n) > 0 for n >= 9).
%D A124478 Pierre Dusart, Autour de la fonction qui compte le nombre de nombres premiers, Th\`ese, Universit\'e de Limoges, France, 1998 [available at http://www.unilim.fr/laco/theses/1998/T1998_01.pdf] see Section 1.9.
%D A124478 M. Hassani, A Remark on the Mandl's Inequality, http://arxiv.org/abs/math/0606765.
%D A124478 Rosser, J. Barkley; and Schoenfeld, Lowell; Sharper bounds for the Chebyshev functions theta(x) and psi(x), Math. Comp. 29 (1975), 243-269.
%K A124478 sign,new
%O A124478 1,1
%A A124478 njas, Dec 17 2006

I would like to get more sequences of this type (I have not gone through 
the Mitrinovic et al., Handbook in a systematic way).

Neil Sloane