Alternate forms of $ \sum\limits_{n=2}^\infty \text P(n)=\sum\limits_{p\text{ prime}}\frac1{p(p-1)} $ with the prime zeta function. We know that: $$\sum_{n=2}^\infty (\zeta(n)-1)=1$$ but what about with the Prime Zeta function $\text P(s)$?:
$$\sum_{n=2}^\infty \text P(n)=0.77315666904979…$$
Now interchange the sum with the prime numbers $ p_k$
$$\sum_{n=2}^\infty \sum_{m=1}^\infty \frac1{p_m^n}= \sum_{m=1}^\infty\sum_{n=2}^\infty \frac1{p_m^n}=\sum_{n=1}^\infty \frac1{p_n(p_n-1)}$$
which gives the same decimal. There are also many prime-related constants. Another form given by @reuns is with the Euler Phi and Mobius functions :
$$\sum_{n=2}^\infty \text P(n)=\sum_{n=2}^\infty \frac{\phi(n)\ln(\zeta(n))}n-\sum_{n=2}^\infty \frac{\mu(n)\ln(\zeta(n))}n= \sum_{n=2}^\infty \frac{\phi(n)\ln(\zeta(n))}n-\text C= \gamma-\text B_1+ \sum_{n=2}^\infty \frac{\phi(n)\ln(\zeta(n))}n $$
where $\text P(s)=-\ln(\epsilon)+\text C+O(\epsilon),\epsilon>0$ as seen in Formulas 3 throught 5 here on MathWorld and $\text C=\text B_1-\gamma$ with the Merten’s constant $\text B_1$ and Euler Mascheroni constant. It also gives the same decimal. The Euler phi sum is similar to @Steven Clark’s formula ($14$) here.
If one read further in the Merten’s constant article, then the amount of prime factors average deviation constant $\text B_2$ appears giving:
$$\sum_{n=2}^\infty \text P(n)=\sum_{p\text{ prime}}\frac1{p(p-1)}=\text B_2-\text B_1= 0.77315666904979… $$
Now that a “closed form” has been found, are there any alternate forms of the $0.77315666904979…\,$ constant in terms of special functions, integrals, manipulated sums et cetera?
 A: I have a formally relative simple answer. Wolfram Alpha is very fluent in numbers. Simply enter the number of your interest in the search/input field of Wolfram Alpha and have a look at the result.
The answer offers for example a continued fraction series for the truncated number. But Mathematica can do infinity numbers.
So continued fraction approximation for $n=20$:
$0.77315666904979512786 \approx  [0; 1, 3, 2, 2, 4, 2, 2, 15, 12, 1, 19, 1, 9, 1, 5, 1, 1, 5, 1, 1]$
Wolfram Alpha suggests some closed forms that are more or less good guesses. This is number representation theory.
Have a look at 0.77315666904979512786.  The non-search result is A136141. I hope that helps.
For the given function $C(x)$ I am able to give many other representations.
$C(x)=\sum_{p=1}^{p\leq x}\frac{log(p)}{p}$.
$\text{StieltjesGamma}[1]-\text{StieltjesGamma}[1,1+(p\leq x)]$
The minuent is the generalized StieltjesGamma function.
In traditional form:
$\gamma _ 1-\gamma _ 1((p\leq x)+1)$.
And with that a whole bunch of different but not so new representations are available.
A: This answer contains more information based on chapter 22 of "An Introduction to the Theory of Numbers" by Hardy and Wright which hopefully provides a bit more insight.
Let
$$C(x)=\sum\limits_{p\le x} \frac{\log(p)}{p}\tag{1}$$
where $p\in\mathbb{P}$ and
$$\tau(x)=C(x)-\log(x)=O(1)\tag{2}$$
then
$$\sum\limits_{p\le x} \frac{1}{p}=\frac{C(x)}{\log (x)}+\int\limits_2^x\frac{C(t)}{t \log^2(t)}\,dt=1+\frac{\tau (x)}{\log (x)}+\int_2^x \frac{1}{t \log(t)}\,dt+\int_2^x \frac{\tau (t)}{t \log ^2(t)} \, dt$$ $$=\log(\log(x))+B_1+E(x)\tag{3}$$
where
$$B_1=1-\log(\log(2))+\int_2^\infty \frac{\tau(t)}{t \log^2(t)}\,dt\tag{4}$$
and
$$E(x)=\frac{\tau(x)}{\log(x)}-\int\limits_x^\infty \frac{\tau(t)}{t \log^2(t)}\,dt=O\left(\frac{1}{\log(x)}\right)\tag{5}$$
so
$$\sum\limits_{p\le x} \frac{1}{p}=\log(\log(x))+B_1+o(1)\tag{6}$$
Also
$$B_1=\gamma+\sum\limits_p \left(\log\left(1-\frac{1}{p}\right)+\frac{1}{p}\right)\tag{7}$$
so
$$\sum_{n=2}^\infty P(n)=\sum\limits_{n=2}^\infty \frac{\phi(n)}{n} \log(\zeta(n))+\gamma-B_1\tag{8}=\sum\limits_{n=2}^\infty \frac{\phi(n)}{n} \log(\zeta(n))-\sum\limits_p \left(\log\left(1-\frac{1}{p}\right)+\frac{1}{p}\right)$$
A couple of other relationships are
$$\sum\limits_{n=1}^x \nu(n)=x \log(\log(x))+B_1 x+o(x)\tag{9}$$
$$\sum\limits_{n=1}^x \Omega(n)=x \log(\log(x))+B_2 x+o(x)\tag{10}$$
where $\nu(n)$ and $\Omega(n)$ are the number of distinct and non-distinct primes dividing $n$.
