Strong Achilles Numbers (noch nicht übersetzt)

A positive integer n is powerful if p² 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 = 2⁵·3³ and 1800 = 2³·3²·5².

We shall call a positive integer S a Strong Achilles number if both S and φ(S) are Achilles numbers.¹
For example, 864 is a Strong Achilles number: φ(864) = 288 = 2⁵·3². However, 1800 isn't a Strong Achilles number because: φ(1800) = 480 = 2⁵·3¹·5¹.

There are 7 Strong Achilles numbers below 10⁴ and 656 below 10⁸.

How many Strong Achilles numbers are there below 10¹⁸?

¹ φ denotes Euler's totient function.