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$\displaystyle\sum _{ n=2 }^{ \infty }{ \dfrac { \zeta (n)-1 }{ n+2 } } =\dfrac { C }{ H } -I\ln { A } -\dfrac { C_{ 1 }\gamma ^{ K } }{ P } -\dfrac { E\ln { (A_{ 1 }\pi ) } }{ S }$

The equation holds true, where $A$ denotes the Glaisher Kinkelin constant and $\gamma$ denotes the Euler-Mascheroni constant .

And $C,H,I,C_1,K,P,E,A_1,S$ are positive integers with $\gcd(C,H)=\gcd(C_1,P)=\gcd(E,S)=1$ .

Find $C+H+I+C_1+K+P+E+A+S$ .