Vyxal, 9 bytes

ʀEĖeæa)ȯt

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

Explained

ʀEĖeæa)ȯt # 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
       t  # last number in that list

Vyxal, 9 8 bytes

ʀEĖeæa)ȯ

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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

Vyxal, 9 bytes

ʀEĖeæa)ȯt

Try it Online!

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

Vyxal, 9 bytes

ʀEĖeæa)ȯt

Try it Online!

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

Explained

ʀEĖeæa)ȯt # 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
       t  # last number in that list

Vyxal, 9 bytes

ʀEĖeæa)ȯt

Try it Online!

Times out for numbers larger than 20.

Vyxal, 9 bytes

ʀEĖeæa)ȯt

Try it Online!

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