Series of exponential prime reciprocals

[YvesSch]

Sum of reciprocal of some base (I just chose e as example) to prime power?

Ʃ \frac{1}{e^{p}} = \frac{1}{e^2}+\frac{1}{e^3}+\frac{1}{e^5}+\frac{1}{e^7}+\frac{1}{e^{11}}+\frac{1}{e^{13}}+\frac{1}{e^{17}}+...
p\inP

Brute force simulation gives me
~0.19279118970439518
Is there an elementary, non-transient solution?

 

[Norwegian]

If you replace e by 2, you get what some people call the prime constant. It encodes all the primes, but as far as I can tell, no other interesting relations involving this number has been found. Also, it may still be open whether this number is algebraic or transcendental.