sequences from inequalities

[N. J. A. Sloane]

Jud said:
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>From: Jud McCranie <j.mccranie at adelphia.net>
>Subject: Re: sequences from inequalities
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>At 05:53 PM 8/30/2004, N. J. A. Sloane wrote:
>
> >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.
>
>%N A097604 Floor( phi(n)*sqrt*2n) ) - n.
>...
>>%C A097604 This is probably known to be always >= 0 - reference?
>
>"Elementary Number Theory", 4th edition, by David Burton, problem 7a in 
>section 7.2 has the equivalent of n/phi(n) <= 2*sqrt(n) - not 
>sqrt(2n).  That's probably where I got that inequality, but wrote it down 
>incorrectly.
>
>

Me:  In my last message I mentioned the Mitrinovic
H'book of No. Thy.  In fact the very first inequlatity
in the book is

phi(n) >= sqrt(n),for n not 2 or 6 [giving sequence A079530]
which of course implies x/phi(x) <= sqrt( 2x)

NJAS