Given a composite number with factorization
where the are all distinct primes, and the are positive integers (and also distinct, always ), and is the number of distinct prime factors function, then is a squarefull number, and if it is also the case that there is no solution to in integers (with and ), then is called an Achilles number.
So, for example, 1323, being , is divisible by the squares of both 3 and 7, but its square root is approximately 36.373, its cubic root is about 10.9779, its fourth root is about 6.031, its fifth root is about 4.21, its sixth root is 3.313, seventh root about 2.792, etc.
According to Greek legend, Achilles was a powerful, seemingly invincible warrior who fought in the Trojan War. In some versions of the story, his mother Thetis held baby Achilles by the left heel and dipped him into the Styx river, making him invulnerable to attack anywhere on his body except the one heel. Thus, Achilles was very powerful, but not perfect because of his fatal weak spot. Likewise, Achilles numbers are powerfull but they are not perfect powers.