Let   $f\left(x\right)={(x^2+3x+2)}^{\cos \left(\pi x\right)}$. Find the sum of all positives integers n for which  $\left|\sum _{k=1}^n{\log }_{10}f\left(k\right)\right|=1.$

Note that $\cos \left(\pi x\right)$ is -1 when $x$ is odd and 1
When $x$ is even. Also note that  $x^2+3x+2=(x+1)(x+2)$
For all  $x$. Therefore
${\log }_{10}f\left(x\right)={\log }_{10}(x+1)+{\log }_{10}(x+2)$ if $x$ is even
${\log }_{10}f\left(x\right)=-{\log }_{10}(x+1)-{\log }_{10}(x+2)$ if $x$ is odd
Because of this,$\sum _{k=1}^n{\log }_{10}f\left(k\right)$  is a telescoping 
series of logs, and we have
$\sum _{k=1}^n{\log }_{10}\frac{n+2}{2}$ if n is even
$\sum _{k=1}^n{\log }_{10}f\left(k\right)={\log }_{10}(n+2)-{\log }_{10}2$
$={\log }_{10}2(n+2)$ if n is odd 
Setting each of the above quantities to 1 and -1 
And solving for n, we get possible values of  $n=3andn=18$
So our desired answer is  $3+18=21$