Achilles Numbers, Powerful Numbers and Perfect Powers

Today I turned 24500 days old and an analysis of this number revealed that it was an Achilles number in the OEIS listings, specifically OEIS A203663: Achilles number whose double is also an Achilles number. The initial members of this sequence are listed as:

432, 972, 1944, 2000, 2700, 3456, 4500, 5292, 5400, 5488, 8748, 9000, 10584, 10800, 12348, 12500, 13068, 15552, 16000, 17496, 18000, 18252, 21168, 21296, 21600, 24300, 24500, 24696, 25000, 26136 

I'd not heard of an Achilles number before and so I investigated. Wikipedia provides this definition:

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 factorisation. 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.

Now 24500 factorises to $2^2 \times 5^3 \times 7^2$ and is thus powerful but not a perfect power. So it is an Achilles number. Multiplication by 2 does not change this because 49000 factorises to $2^3 \times 5^3 \times 7^2$ and again powerful but not a perfect power.

Here is the Wikipedia definition of a perfect power:

A perfect power is a positive integer that can be expressed as an integer power of another positive integer. More formally, $n$ is a perfect power if there exist natural numbers $m > 1$, and $k > 1$ such that $m^k = n$. In this case, $n$ may be called a perfect $k$-th power. If $k = 2$ or $k = 3$, then $n$ is called a perfect square or perfect cube, respectively. Sometimes 1 is also considered a perfect power ($1^k = 1$ for any $k$).