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 \(k\)).