Nearly Gorenstein vs almost Gorenstein affine monomial curves

Abstract: We extend some results on almost Gorenstein affine monomial curves to the nearly Gorenstein case. In particular, we prove that the Cohen-Macaulay type of a nearly Gorenstein monomial curve in $\mathbb{A}^4$ is at most $3$, answering a question of Stamate in this particular case. Moreover, we prove that, if $\mathcal C$ is a nearly Gorenstein affine monomial curve which is not Gorenstein and $n_1, \dots, n_{\nu}$ are the minimal generators of the associated numerical semigroup, the elements of ${n_1, \dots, \widehat{n_i}, \dots, n_{\nu}}$ are relatively coprime for every $i$.