# Problem 302: Strong Achilles Numbers
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.