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Are there infinitely many numbers $abc...z$ with $d$ digits such that $a^k + b^{k+1} + c^{k+2} + \dots + z^{k+d-1} = abc...z$ for a positive integer k? For k=1 the largest is $12157692622039623539$, and there are 10 non-trivial solutions. For k=2 however I only know of $43 = 4^2 + 3^3$ and $63 = 6^2 + 3^3$. I haven't been able to find any solutions for higher values of k.

Are there any nontrivial solutions for $k \ge 3$?

Are there any solutions for k=2 besides 43 and 63?

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For high enough $k$ / $d$, you can't have digits larger than 1 in certain positions.

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