Is $\sum\frac1{p^{1+ 1/p}}$ divergent?

Is $\displaystyle\sum\frac1{p^{1+ 1/p}}$ divergent? How can we prove that it is divergent or convergent in analytic number theory?
I know what bound of the n-th prime number is, and that its order is $n\log(n)$.
Maybe we can use the divergence of $\displaystyle\sum\frac1{n^{1+ 1/n}}$ to show that. I’m not sure that $\displaystyle\sum\frac1{n^{1+ 1/n}}$ is divergent, but I think it is.
So would you please help me with this ? Can you help me in finding a proof for it ?
Thank you very much, friends.

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