Achilles number

An Achilles number is a number that is powerful but not a perfect power. A positive integer $n$ is a powerful number if, for every prime factor $p$ of $n$ , $p 2$ 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.

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