On powers that are sums of consecutive like powers

Let $k \ge 2$ be even, and let $r$ be a non-zero integer. We show that for almost all $d \ge 2$ (in the sense of natural density), the equation $x^k+(x+r)^k+\cdots+(x+(d-1)r)^k=y^n, \qquad x,~y,~n \in \mathbb{Z}, \qquad n \ge 2,$ has no solutions.

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