2021-11 Interesting perfect cubes

Determine if there exist infinitely many perfect cubes such that the sum of the decimal digits coincides with the cube root. If there are only finitely many, how many are there? 

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About Hyungryul

2003.3-2009.8 KAIST, Undergraduate student in Mathematics 2009.8-2014.8 Cornell University, PhD student in Mathematics 2014.9-2017.2 University of Bonn, Postdoc 2017.3-2021.2. KAIST, Assistant Professor 2021.3-Present. KAIST, Associate Professor