## Strong Achilles Numbers

### Problem 302

Published on Saturday, 18th September 2010, 07:00 pm; Solved by 477; Difficulty rating: 60%
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}·3^{3} and 1800 = 2^{3}·3^{2}·5^{2}.

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 = 2^{5}·3^{2}. However, 1800 isn't a Strong Achilles number because: φ(1800) = 480 = 2^{5}·3^{1}·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} φ denotes **Euler's totient function**.