Achilles number

An Achilles number is a number that is powerful but not a perfect power.^[1] A positive integer n is a powerful number if, for every prime factor p of n, p² is also a divisor. In other words, every prime factor appears at least squared in the factorization. All Achilles numbers are powerful. However, not all powerful numbers are Achilles numbers: only those that cannot be represented as m^k, where m and k are positive integers greater than 1.

Achilles numbers were named by Henry Bottomley after Achilles, a hero of the Trojan war, who was also powerful but imperfect. Strong Achilles numbers are Achilles numbers whose Euler totients are also Achilles numbers; the smallest are 500 and 864.^[2]