An Achilles number is a positive integer that is powerful (every prime exponent ≥ 2) but is not a perfect power (i.e., not of the form \(m^k\) for any integer \(m\) and any \(k > 1\)).
The first Achilles numbers: 72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125.
The name evokes Achilles' heel: these numbers are "strong" (powerful) yet have a vulnerability (not a perfect power). Equivalently, the GCD of the exponents in the prime factorization is exactly 1, even though every individual exponent is at least 2.
Example: \(72 = 2^3 \cdot 3^2\) — both exponents ≥ 2 (powerful), but \(\gcd(3, 2) = 1\) (so not a perfect power). Achilles.