##Seriously, 25 bytes

,r`PªD;⌐k`M┬`π`Mi│g;)@\)\

Outputs a\nb (\n is a newline). Large inputs will take a long time (and might fail due to running out of memory) because prime generation is pretty slow.

Try it online!

Explanation:

,r`PªD;⌐k`M┬`π`Mi│g;)@\)\
,r                         push range(input)
  `PªD;⌐k`M                map:
   P                         k'th prime
    ª                        square
     D                       decrement
      ;                      dupe
       ⌐                     add 2 (results in P_k + 1)
        k                    push to list
           ┬               transpose
            `π`M           map product
                i│         flatten, duplicate stack
                  g;)      push two copies of gcd, move one to bottom of stack
                     @\    reduce denominator
                       )\  reduce numerator

Seriously, 25 bytes

,r`PªD;⌐k`M┬`π`Mi│g;)@\)\

Outputs a\nb (\n is a newline). Large inputs will take a long time (and might fail due to running out of memory) because prime generation is pretty slow.

Try it online!

Explanation:

,r`PªD;⌐k`M┬`π`Mi│g;)@\)\
,r                         push range(input)
  `PªD;⌐k`M                map:
   P                         k'th prime
    ª                        square
     D                       decrement
      ;                      dupe
       ⌐                     add 2 (results in P_k + 1)
        k                    push to list
           ┬               transpose
            `π`M           map product
                i│         flatten, duplicate stack
                  g;)      push two copies of gcd, move one to bottom of stack
                     @\    reduce denominator
                       )\  reduce numerator