Achilles number

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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 divisor or factor $p$ of $n$ , $p 2$ is also a divisor. In other words, every prime factor appears squared. 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, put laconically, are powerful but imperfect (as in an not a perfect power): not entirely unlike Achilles, a hero of the Trojan war.

[edit] Sequence of Achilles numbers

The Achilles numbers between 1 and 5000 are:

72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000 (sequence A052486 in OEIS)

[edit] Examples

108 is a powerful number. Its prime factorization is $2^2 \cdot 3^3$ , and thus its prime factors are 2 and 3. Both $22 = 4$ and $32 = 9$ are divisors of 108. However, 108 cannot be represented as $m k$ , where $m$ and $k$ are positive integers greater than 1, so 108 is an Achilles number.

Finally, 784 is not an Achilles number. It is a powerful number, because not only are 2 and 7 its only prime factors, but also are $22 = 4$ and $72 = 49$ divisors of it. Nonetheless, it is a perfect power:

$784=2^4 \cdot 7^2 = (2^2)^2 \cdot 7^2 = (2^2 \cdot 7)^2 = 28^2$