Nearly Gorenstein and almost symmetric properties in shifted numerical semigroups

Given the integers $0<r_1<\dots<r_k$, we consider the shifted family of semigroups $M_n=\langle n, n+r_1,\dots, n+r_k\rangle$, where $n>0$.

For sufficiently large $n$, we prove that if $M_n$ is nearly Gorenstein or almost symmetric, then so is $M_{n+r_k}$.

A key ingredient is to relate the pseudo-Frobenius elements of $M_n$ and $M_{n+r_k}$, correcting a wrong claim in the literature.

Moreover, we derive explicit formulas for the Frobenius and pseudo-Frobenius numbers of $M_{n+r_k}$.