On the order of magnitude of certain integer sequences

Let p be a prime number, and let S be the numerical semigroup generated by the prime numbers not less than p. We compare the orders of magnitude of some invariants of S with each other, e. g., the biggest atom u of S with p itself: By Harald Helfgott (arXiv:1312.7748 [math.NT]), every odd integer N greater than five can be written as the sum of three prime numbers. There is numerical evidence suggesting that the summands of N always can be chosen between N6 and N2. This would imply that u is less than $6p$.