When does $x^3+y^3=kz^2$?

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

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

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

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

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

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

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