Why are there only 7 numbers like this? Proving it with Python code, with help from ChatGPT

This viral trick has tens of millions of views, and leads to a natural question: which numbers have cube roots exactly equal to the number's digit sum? We can tackle this question mathematically, with an assist from ChatGPT and Python code, and figure out all possible values!

Sum of cube root digits short video
   • Trick!  

Dudeney number
https://en.wikipedia.org/wiki/Dudeney...

Proof of Dudeney numbers
http://blog.hostilefork.com/six-duden...

OEIS Dudeney numbers
https://oeis.org/A061209

ChatGPT
https://chat.openai.com/

Python online compiler
https://www.programiz.com/python-prog...

Remove space Python print
https://discuss.python.org/t/how-to-r...

Math StackExchange
https://math.stackexchange.com/questi...

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