TITLE: Does $\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ converge to the golden ratio?
QUESTION [15 upvotes]: The sum $\displaystyle\sum\limits_{n=2}^{\infty}\frac{1}{n\ln(n)}$ does not converge.
But the sum $\displaystyle\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ where $P_n$ denotes the $n$th prime number appears to be.
Is that correct, and if so, how can we calculate the value of convergence?
Is it possible that this sum converges to the golden ratio ($\dfrac{1+\sqrt{5}}{2}$)?

REPLY [5 votes]: 1.63661632335... 
See http://oeis.org/A137245 and links therein
(also http://en.wikipedia.org/wiki/Prime_zeta_function, scroll down to integral section)