Consider the following series:
$$\sum_{p\in\mathcal{P}}\frac{1}{p^p}$$
where $\mathcal{P}$ is the set of all prime numbers: $\mathcal{P}=\{2,3,5,7,11,13,\ldots\}$. My question is:
Is this a known series? In case it is, what is it's name in literature?
With a simple comparison, we conclude that it is convergent and:
$$0\leq \frac{1}{p^p}\leq \frac{1}{p^2}\Rightarrow 0\leq\sum_{p\in\mathcal{P}}\frac{1}{p^p}\leq\sum_{p\in\mathcal{P}}\frac{1}{p^2}<\sum_{n=2}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}-1\approx 0.64493$$
and for $p\in\{2,3,5,7,11,13\}=\mathcal{P}_{13}$ my calculator outputed:
$$\sum_{p\in\mathcal{P}_{13}}\frac{1}{p^p}\approx 0.287358251$$
But what else can we say about it? Is it a rational number? In case it isn't rational, is it algebraic (and if it is what can we say about the minimal polynomial)?