For the semigroup generated by primes at least p_n, the largest atom u_n satisfies u_n ~ 3 p_n; Wilf's inequality holds for these semigroups, and if the Frobenius number f_n is eventually odd then f_n ~ 3 p_n, which would imply Goldbach for large evens.
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Numerical Semigroups generated by Primes
For the semigroup generated by primes at least p_n, the largest atom u_n satisfies u_n ~ 3 p_n; Wilf's inequality holds for these semigroups, and if the Frobenius number f_n is eventually odd then f_n ~ 3 p_n, which would imply Goldbach for large evens.