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 $\frac N6$ and $\frac N2$. This would imply that $u$ is less than $6p$.