Think twice

In a preliminary step, let us count the number of generators of the cyclic group of order $n$ . If $g$ is a generator, then $g^m$ will be a generator iff $(g^m)^a=g$ for some $a$ iff $ma\equiv 1 \pmod{n}$ iff $1=am+bn$ for some $a,b$ iff $\gcd(m,n)=1$ . Thus there are $\phi(n)$ generators.

Now the primitive roots modulo $n$ are just the generators of the multiplicative group modulo $n$ , by definition. Since that group is cyclic for odd prime powers and its order is $\phi(n)$ , the number of (incongruent) primitive roots is $\phi(\phi(n))$ ... thus "think twice".

In particular, we have $\phi(\phi(7^7))=\phi(6\times 7^6)=2\times6\times 7^5=\boxed{201684}$