Is the sum $$\sum_p \frac{1}{p^p}=0.2873582513062241797364\cdots $$ where $p$ runs over the primes, a transcendental number ?
I don't think that it is a Liouville-number , but I am not sure. However, the sequence $\frac{1}{p^p}$ might decrease fast enough to prove the transcendentality. However I am not sure whether it is even easy to show that the sum is irrational.
Any ideas or perhaps a reference ?