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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.

A sqube is a positive integer $p^2{q}^3$ for distinct prime numbers $p,q$. The ordering matters: $3^2\cdot 2^3=72$, whereas $2^2\cdot 3^3=108$. By unique prime factorization, the exponents identify the ordered pair $p,q$ from the resulting integer.

Every sqube is an Achilles number. It is a powerful number, because both prime exponents are at least two. It is not a perfect power, because the greatest common divisor of the exponents $2,3$ is one. This connects the square-cube construction to two general factorization classes.

A strong Achilles number is a positive integer $n$ for which both $n$ and its Euler totient $\phi (n)$ are Achilles numbers. For example,In each prime factorization, every prime exponent is at least two and their greatest common divisor is one. Thus both numbers are powerful without being perfect powers. The extra condition matters: $72$ is an Achilles number, but $\phi (72)=24=2^{3}3$ is not powerful, so $72$ is not a strong Achilles number.