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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.