When does $x^3+y^3=kz^2$? For which integers $k$ does
$$
x^3+y^3=kz^2
$$
have a solution with $z\ne0$ and $\gcd(x,y)=1$? Is there a technique for counting the number of solutions for a given $k$?

$\qquad\qquad$ Too long for a comment: Positive coprime integers x and y less than $6,000$ :

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad$ $k=1$

$\qquad\qquad\qquad\qquad$ 

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad$ $k=2$

$\qquad\qquad\qquad\qquad$ 

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad$ $k=3$

$\qquad\qquad\qquad\qquad$ 

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad$ $k=4$

$\qquad\qquad\qquad\qquad$ 

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad$ $k=5$

$\qquad\qquad\qquad\qquad$ 
