An Achilles number is a powerful number that is not a perfect power. As such, Achilles numbers never have integers as square roots, cube roots, or any other root despite having a prime factorization entirely of powers. Named by Henry Bottomley for the Greek hero who was powerful but not without flaw. A few examples of Achilles numbers:

72 = 2^{3} ∗ 3^{2} (this is also the smallest one)

2,700 = 2^{2} ∗ 3^{3} ∗ 5^{2}

It's worth noting that not every powerful number whose prime factors have different exponents will be an Achilles number. If all of the exponents themselves have a common factor they will still be a perfect power. For instance:

1,936 = 2^{4} ∗ 11^{2} = 44^{2}

1,728 = 2^{6} ∗ 3^{3} = 12^{3}

The On-Line Encyclopedia of Integer Sequence list for Achilles numbers can be found here.

IRON NODER XIV: THE RETURN OF THE IRON NODER