boring numbers

[Marc LeBrun]

The period 1 continued fractions [a a a...] involve taking the square roots 
of numbers of the form n^2+1.

The square part of n^2+1 seems always to be a prime congruent to 1 mod 
4.  This looks to be A002144, although for n<10000 only the 29 values

   5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149,
   181, 193, 197, 257, 281, 313, 373, 389, 401, 569, 577, 677, and 5741

appear.  If I knew enough modern quadratic residue theory, would I be sure 
that all such primes would eventually appear?  Would I know when each one 
would first appear?  Would I also be able to characterize the sequence of 
square free parts of n^2+1?

Thanks!