What is known about the sum $\sum\frac1{p^p}$ of reciprocals of primes raised to themselves? [duplicate]
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.
Solution 1:
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 ..."
Solution 2:
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.
Solution 3:
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 $\sum_{n=6}$
Solution 4:
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).