1. Note that any number that satisfies must have at least one prime divisor - prove this! -, now try to construct a number which is but not multiple of any of the "finite" number of primes - adapt Euclid's proof so that you build a number congruent to 3 module 4.
2. That series diverges but grows painfully slowly (there's a quote I heard that goes something like: " log log x tends to infinity, but no one has actually observed this in practice" )
A lower bound would be:
We have (since )
Now from Euler's proof you should know that and so with this and (1) we are done.
So now using this we get that roughly for your inequality should hold. - observe how HUGE that is!
Now, finding the smallest for which the inequality holds is totally unreasonable in my opinion.