Let p(1) = 2, p(2) = 3, p(3) = 5,. be the consecutive prime numbers, S-n the numerical semigroup generated by the primes not less than p(n) and u(n) the largest irredundant generator of S-n. We will show, that u(n) similar to 3p(n). Similarly, for the largest integer f(n) not contained in S-n, by computational evidence (https://www.uni-regensburg.de/Fakul taeten/nat_Fak_I/Hellus/table 1.pdf) we suspect that (1) f(n) is an odd number for n = 5 and (2) f(n) similar to 3p n; further (3) 4p(n) > f(n+1) for n >= 1. If f n is odd for large n, then f(n) similar to 3p(n). In case f(n) similar to 3p(n) every large even integer x is the sum of two primes. If 4p(n) > f(n+1) for n >= 1, then the Goldbach conjecture holds true. Further, Wilf's question in Wilf (Am Math Mon 85:562-565, 1978) has a positive answer for the semigroups S-n.