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, p2 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. Powerful nummbers that can be represented as mk, where m and k are positive integers greater than 1 are not Acholies numbers.

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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