A positive integer $n$ is powerful if $p^2$ is a divisor of $n$ for every prime factor $p$ in $n$.

A positive integer $n$ is a perfect power if $n$ can be expressed as a power of another positive integer.

A positive integer $n$ is an Achilles number if $n$ is powerful but not a perfect power. For example, $864$ and $1800$ are Achilles numbers: $864=2^5\cdot 3^3$ and $1800=2^3\cdot 3^2\cdot 5^2$.

We shall call a positive integer $S$ a Strong Achilles number if both $S$ and $\varphi (S)$ are Achilles numbers.¹
For example, $864$ is a Strong Achilles number: $\varphi (864)=288=2^5\cdot 3^2$. However, $1800$ isn't a Strong Achilles number because: $\varphi (1800)=480=2^5\cdot 3^1\cdot 5^1$.

There are $7$ Strong Achilles numbers below ${10}^4$ and $656$ below ${10}^8$.

How many Strong Achilles numbers are there below ${10}^{18}$?

¹ $\varphi$ denotes Euler's totient function.