The probability that two integers picked at random are relatively prime is known to be $1/\zeta{(2)}=6/\pi^2\approx0.607927...$. Generalizing, the probability that $n$ random integers have $\gcd=1$ is $1/\zeta{(n)}$.
What is the probability that two random integers have one (and only one) prime factor in common? I did some calculations and obtained the formula $\displaystyle\frac{P(2)}{\zeta(2)}=0.274933...$, where $P(n)$ is the prime zeta function. In general, I suppose that the probability that $n$ random integers have only one prime factor in common is $\displaystyle\frac{P(n)}{\zeta(n)}$.
I would be interested in having confirmation of these results, and in getting a formal proof. I would also like to obtain a generalization expressing the probability that $n$ integers picked at random have exactly $k$ prime numbers in common.