Minimal genus of a multiple and Frobenius number of a quotient of a numerical semigroup

Given two numerical semigroups $S$ and $T$ and a positive integer $d$, $S$ is said to be one over $d$ of $T$ if $S=\{s \in \mathbb{N} \ | \ ds \in T \}$ and in this case $T$ is called a $d$-fold of $S$. We prove that the minimal genus of the $d$-folds of $S$ is $g + \lceil \frac{(d-1)f}{2} \rceil$, where $g$ and $f$ denote the genus and the Frobenius number of $S$. The case $d=2$ is a problem proposed by Robles-P\'erez, Rosales, and Vasco. Furthermore, we find the minimal genus of the symmetric doubles of $S$ and study the particular case when $S$ is almost symmetric. Finally, we study the Frobenius number of the quotient of some families of numerical semigroups.