Numbers with similar powers
Husk, 10 bytes
§ḟ¤≡ȯÖLgp→

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Explantion
§ḟ       →     Find the first number starting from the input + 1 such that...
        p        The prime factorisation
       g         with equal elements grouped together
    ȯÖL          and sorted by length of groups
  ¤≡             has the same shape as the above applied to the input.