I have a question about numerical semigroups. It is known that if $a, b\in \mathbb{N}$ and $\gcd(a, b)=1$, then the numerical semigroup $\langle a, b \rangle$ has genus $\frac{(a-1)(b-1)}{2}$.
My question is: If $a, b, c \in \mathbb{N} $ are such that $\gcd(a, c)=1$ and $g(H)=\frac{(a-1)(b-1)}{2}$, where $H=\langle a, b, c \rangle$. I can affirm that $\gcd(a, b)=1$ and $H=\langle a, b \rangle$?
I've been trying to prove it to no avail. Also, if it's not valid, I can't find a counterexample.