Observation is important

Given that $A^n=16^{17}A$ , so $A^{n-1}=(2^4)^{17}=2^{68}$ .

Therefore, $n-1$ is a factor of 68,

the sum of all possible value of $n-1$ is the sum of factor of $68(=2^2\times 17^1)$ ,

so the sum of all possible value of $n-1$ is $(2^0+2^1+2^2)(17^0+17^1)=7\times 18=126$

Due to there are $(2+1)(1+1)$ $n$ , therefore the sum of all possible value of $n$ is $126+6=132$