The number of primes not in a numerical semigroup

Abstract

For two coprime positive integers $a$ and $b$,let $\pi^* (a, b)$ be the number of primes that cannot be represented as $au+bv$, where $u$ and $v$ are nonnegative integers. It is clear that $\pi^* (a, b)\le \pi (ab-a-b)$, where $\pi (x)$ denotes the number of primes not exceeding $x$. In this paper, we prove that $\pi^* (a, b)\ge 0.04\pi (ab-a-b)$ and pose following conjecture: $\pi^* (a, b)\ge \frac 12 \pi (ab-a-b)$. This conjecture is confirmed for $1\le a\le 10$.