Given these two computational complexities of 2 algorithms:
$\exp(O(\sqrt{\log n \log \log n}))$
$O(\sqrt{\exp n} / \log{ \sqrt{ \exp n} })$
where I imagine the first one goes to infinity slower than the second one, what could be a "simple" function $F(n)$ such that, when multiplied by the second complexity, makes it go to infinity with "same speed" of the first one ? (Lets say, more precisely, so that the ratio of the functions be asymptotically bounded)
[Note that the first expression comes essentially from Wikipedia: http://en.wikipedia.org/wiki/Integer_factorization#General-purpose, while the second would like to represent a simple process of division by the primes present before the square root of the number being factored]