We show that for almost all $n\in \mathbf{N}$, the inequality$$|p_1+p_2-\exp \left(\right(\log n{)}^{\gamma }\left)\right|\<1$$has solutions with odd prime numbers $p_1$ and $p_2$, provided $1\<\gamma \<\frac{3}{2}$. Moreover, we give a rather sharp bound for the exceptional set.This result provides almost-all results for Goldbach numbers in sequences rather thinner than the values taken by any polynomial.