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.
强阿喀琉斯数
我们称正整数n是强大的（似幂的），如果对于n的每一个质因数p，p2都是n的约数。
我们称正整数n是完美的（完全幂的），如果n可以表达成另一个正整数的幂。
我们称正整数n是阿喀琉斯数，如果n是强大的但不是完美的。例如，864和1800都是阿喀琉斯数：864 = 25·33，而1800 = 23·32·52。
我们称正整数S是强阿喀琉斯数，如果S和φ(S)都是阿喀琉斯数。1
例如，864是强阿喀琉斯数：φ(864) = 288 = 25·32。然而，1800不是强阿喀琉斯数：φ(1800) = 480 = 25·31·51。
一共有7个小于104的强阿喀琉斯数，656个小于108的强阿喀琉斯数。
有多少个小于1018的强阿喀琉斯数？
1 φ表示欧拉总计函数。