Consider the series: $$\sum_{i = 1}^{\infty}\frac{1}{\mathrm{Prime}(i)^{\mathrm{Prime}(i)}}$$ where Prime(i) denotes the i-th prime.
Proving the convergence of this series is trivial but finding the value to which converges has defied me so far. Mathematica says this series converges to ≈0.287358.
I tried Googling about this series and found very little information about this series.
My questions are:
What does this series converge to? Does this series arise in any context and are there interesting trivia to be known about this series? Is it irrational, transcendental? Does it have a name?