Primes of Ulam's Spiral
Pyth, 20 bytes

f}TPTsuC+R=hZ_GtyQ]]


Try it online: Demonstration

This code generates the fully Ulam's spiral, connects all lines and filters for primes. 

Explanation:

f}TPTsuC+R=hZ_GtyQ]]   implicit: Z = 0
      u           ]]   reduce, start with G = [[]]
               tyQ     for H in [0, 1, ..., 2*input-2] do:
             _G           reverse the order of the lines
        +R=hZ             append Z + 1 at the end of each line, 
                          updating Z each time with the new value Z + 1
       C                  update G with the transposed of ^
                       this gives the Ulam's spiral
     s                 combine all lines to a big list of numbers
f                      filter for numbers T, which satisfy:
 }TPT                    T appears in the prime-factorization of T
                         (&lt;==&gt; T is prime)