What are some integers co-prime with some general $n$?

The answer is all powers of $2$. Remember that $n$ is co-prime with $n-1$ and $n+1$. Therefore, the arithmetic sequence must have a common difference of $2$. Since $1$ is co-prime with every positive integer, the sequence must be $1, 3, 5, 7, \dots$. The only numbers satisfying this are powers of $2$.