Vyxal, 9 8 bytes

ʀEĖeæa)ȯ

Try it Online!

Times out for numbers larger than 20. 1-indexed.

-1 thanks to @Shaggy

Explained

ʀEĖeæa)ȯ  # takes a single integer n
      )ȯ  # first n numbers where:
ʀ         #   the range [0, that number]
 EĖe      #   to the power of 1/(2 to the power of each number in that range) 
    æa    #   contains a prime number