 Achilles number

An Achilles number is a powerful number greater than one that is not a perfect power. In its prime factorizationevery prime exponent satisfies $e_i\ge 2$, and $\gcd (e_1,\dots ,e_k)=1$. The first condition is precisely the definition of a powerful number. For the second, $n=m^r$ with integers $m,r>1$ exactly when $r$ divides every prime exponent; hence a greatest common divisor greater than one is precisely the obstruction.

For example, $72=2^3{3}^2$ is an Achilles number because $\gcd (3,2)=1$. By contrast, $144=2^4{3}^2=12^2$ is powerful but is a perfect square, so it is not an Achilles number.

An Achilles number is a positive integer that is a powerful number but not a perfect power. A powerful number is defined as a number \( n \) such that in its prime factorization, every prime number \( p \) appears with an exponent of at least 2. In contrast, a perfect power is a number of the form \( m^k \) where \( m \) and \( k \) are positive integers and \( k \geq 2 \).