The maximum of the complementary of a semigroup with restricted conditions

Abstract

We consider the set $$\mathcal{A}=\left\{10\cdot a+11\cdot b|\gcd (a,b)=1,a\ge 1,b\ge 2a+1\right\}.$$ We will prove that $\mathcal{A}$ is unbounded and that there exists a natural number $M\notin \mathcal{A}$ for which $$\left\{M+m:m\ge 1,m\in \mathbb{N}\right\}\subset \mathcal{A}.$$ Indeed, such number is $M=1674$.

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