sequences from inequalities

[N. J. A. Sloane]

Alonso / Jud said:
>>Does anyone know what the maximum possible value of x/phi(x) can be
>>(where phi is Euler's totient function)? Is there a theorem in regards
>>to this?
>>
>> From some playing around with Mathematica it seems to me that the
>>value can be made as large as one wants by choosing a sufficiently
>>large highly composite number, but I'm wondering if x/phi(x) is
>>bounded by some property of x, such as its square root.
>
>There is a minimum for that ratio.  There is another theorem that says that 
>x/phi(x) <= sqrt( 2x), so the ratio must be unbounded.
>

Me:  let me remind everyone that the correct thing for seqfans to do 
here is to convert these inequalites into sequences, and then send them
in to the OEIS if they are not there!

Example:  It is suggested that  x/phi(x) <= sqrt( 2x).  If this is true
- and even if it isn't - we get a sequence:

%I A097604
%S A097604 0,0,1,1,7,0,15,8,16,7,35,7,48,17,28,29,76,18,91,30,56,44,126,31,116,60,
%T A097604 105,61,184,31,205,96,129,97,165,65,272,118,172,103,321,67,346,143,182,
%U A097604 165,398,108,366,150,272,192,482,133,364,197,327,243,571,115,601,272,341
%N A097604 Floor( phi(n)*sqrt*2n) ) - n.
%C A097604 This is probably known to be always >= 0 - reference?
%O A097604 1,5
%K A097604 nonn
%A A097604 njas, based on emails from Alonso Del Arte (alonso.delarte(AT)gmail.com) and Jud McCranie (j.mccranie(AT)adelphia.net), Aug 30 2004

which in this case is new.

BTW, Mitrinovic's Handbook of Number Theory is an excellent
reference for this kind of thing. I didn't check it to see if
this is in there.  I've taken a number of sequences based on inequalties
in that reference, but I haven't gone through it 
systematically.

NJAS