Value of $\sum 1/p^p$ A very simple question, but I can't seem to find anything relating to it :
Is there any research, are there any results that have focused on or given insight on 
$\sum 1/p^p$, ${p \in \mathbb P}$ ?
A very basic series, converges extremely fast, its value is around .29. What more can there be said about it ?
From what little I know about more advanced number theory, similar sequences (I can think of a few similar ones that I can't find any relevant research or results about) can be very non-trivial to compute or to analyse.
 A: What more can there be said about it ?
Essentially nothing. Related series are


*

*Sophomore's constant $\displaystyle C_s=\sum_{n=1}^{\infty}\frac{1}{n^n}$.

*Prime zeta values $\displaystyle P(k)=\sum_{p\in\mathbb P}\frac{1}{p^k}$, with $k\in\mathbb N_{\ge 2}$.
No closed form expressions for these constants are known so far.
A: This is OEIS A094289, where they have no information except computations of the value.  This suggests the answer "no" to the question "is there any research ..."
A: it may not be elegant but as an idea $\sum = \sum_{n=1}^{n=5}1/{p^p} + \sum_{n=6}1/{p^p}$ 
and as per this leverage approximation / boundaries 
$\log{n} + \log{\log{n}} - 1 < \frac{p_n}{n} < \log{n} + \log{\log{n}}$  for $n \geq 6$
and then look at convergence of $\sum_{n=6}$
A: Other than its rational approximation
$$
\frac{5226294}{18187381},
$$
not much is known about this constant. In particular, its irrationality/transcendentality seems to be out of reach with current technology. However, it can be related to similar constants, such as:


*

*The prime zeta function:
$$
P(k)=\sum_{p\in\mathbb P}\frac{1}{p^k}, \ \ Re(k)>1,
$$

*$n$-ary representations of the Prime Constant:
$$
f(k)=\sum_{p\in\mathbb P}\frac{1}{k^p},
$$
(noting that $f(2)$ is the Prime Constant $\rho=0.414682509\ldots$),


*

*Sophomore's constant:


$$
C_s=\sum_{n=1}^{\infty}\frac{1}{n^n}.
$$
As pointed out by others, there is more information on the OEIS website (A094289).
