(In reply to

New solutions not relatively prime by Steve Herman)

I would have been shocked if my solution did find all possible solutions. I had recently solved Taking the Fifth, in which Broll asks about a 'nontrivial' solution to a^3+b^3=c^5 (solutions where a,b,c are not based off of a solution to a'^3+b'^3=c'^2).

Now knowing 183, 201, 219 are nontrivial solutions for a^3+b^3=c^4, I went to the OEIS and found

A051387 "Numbers whose 4th power is the sum of two positive cubes in a nontrivial way".