$2^46^118^1=3^34^116^1(=2^63^3)$.
$2\cdot6\cdot18=216>192=3\cdot4\cdot16$.
$(1/2)(5/6)(17/18)=85/216<15/32=(2/3)(3/4)(15/16)$.
Concerning the question about the phi-function, for $$n=2^{28}3^{16}=4\times6\times8\times9\times12\times16\times18\times24\times27\times32\times36\times48\times54$$ we have $${\phi(n)\over n}={1\over3}=.333\dots>.32956\dots={3\over4}\cdot{5\over6}\cdot\cdots\cdot{53\over54}$$