Formula for consecutive residue of primitive modulo n. \begin{align*}
3^0 \equiv 1\mod 7\\
3^1 \equiv 3\mod 7\\
3^2 \equiv 2\mod 7\\
3^3 \equiv 6\mod 7\\
3^4 \equiv 4\mod 7\\
3^5 \equiv 5\mod 7\\
3^6 \equiv 1\mod 7\\
3^7 \equiv 3\mod 7\\
\end{align*}
Now just focusing on 1, 3, 2, 6, 4, 5, 1....
How to devise a formula to find the next number.
Like if 2 is given how to find 6 or if 4 is given how to find 5?
I am looking for an explicit function.

You just need to multiply by $3$ and take the residue mod $7$. This is because $3^j$ mod $7$ is the same as $3^{j-1}\cdot 3$ mod $7$.
