My F function is defined as rad((2*s(x)*s(x-1))^2+1). The rad function is the squarefree part of whatever is in the brackets, and the s function is the OEIS A000522, defined by s(0)=1 and s(n+1)=n*s(n)+1. So I have found factors for all n=1 to 104, except 92, and completely factored (1,2,3...32,33,34,35),37,41,43,73,75,98,100 where bracketed numbers are consecutive. For a copy of the factorization status, click here.

I used trial division, rho, p-1, ECM and GNFS to find the factors.