Achilles numbers

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An Achilles number is a positive integer 

n

that is

• powerful, i.e. divisible by the square of its radical,
• but imperfect, i.e. not a perfect power.

Achilles numbers are named after Achilles, a hero of the Trojan war, who was also powerful but imperfect.

Thus an Achilles number 

n

is

$n=\prod _{i=1}^{\omega (n)}{p_{i}}^{\alpha _{i}},\quad \alpha _{i}\geq 2,\gcd(\alpha _{1},\ldots ,\alpha _{\omega (n)})=1,$

where 

ω (n)

is the number of distinct prime factors of 

n

.

A052486 Achilles numbers (...).

{72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, ...}

See also

• Prime signatures