Strong Achilles Numbers
Problem 302Published on Saturday, 18th September 2010, 07:00 pm; Solved by 487; Difficulty rating: 60%
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.