Max value of x/phi(x)

[Jud McCranie]

At 04:59 PM 8/30/2004, Alonso Del Arte wrote:
>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.

There are several sequences dealing with "highly composite" - enter that 
phrase.

Ref: Bach and Shallit, Algorithmic Number Theory, vol 1, theorem 8.8.7 for 
the minimum.  I wrote the other in the margin, but I don't know a reference.