(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".