Probably this has nothing to do with prime numbers, I just experimented a little bit with it and wanted to share it, in case someone has an idea.
Let
$$p_n := n\text{-th prime number , }[a,b]:= \frac{ab}{\gcd(a,b)^2}$$
Then (?) for every natural number $n$ the matrix
$$G_n := \left (\frac{p_{[a,b]}^{[a,b]}}{p_{[a,b]+1}^{[a,b]+1}} \right )_{1\le a,b \le n}$$
is positive definite?