Conjecture: For any n>0, we have a(n) <= n*(n+1), and the Galois group of SF_n(x) = sum_{k=0}^n A005117(k+1)*x^{n-k} over the rationals is isomorphic to the symmetric group S_n.
For another related conjecture, see the author's comment on A005117.
a(4)=7 since SF_4(x)=x^4+2x^3+3x^2+5x+6 is irreducible modulo 7 but reducible modulo any of 2, 3, 5. It is easy to check that SF_4(x)==(x-2)*(x^3-x^2+x+2) (mod 5).
