So, I picked it by choosing 12^3 + 2. I wanted to start with something easily factored (power of 12) and not too small, but not interesting because of its factorization. And I wanted it to be even, since odd numbers tend to have funny quirks. So a simple +2.
What’s amusing is that I did this before looking at the Wikipedia page. According to that, the original number that inspired this paradox was 1729. So just by happenstance, I picked one integer higher than the original “uninteresting” number. Which I guess arguably makes 1730 interesting now (though admittedly not for mathematical reasons).
Four is bad luck in Japanese superstition, because its pronunciation (‘shi’) is the same as the word for ‘death’.