A Note on Generalized Repunit Numerical Semigroups

Let \(A=(a_1, a_2, ..., a_n)\) be relative prime positive integers with \(a_i\geq 2\). The Frobenius number \(F(A)\) is the largest integer not belonging to the numerical semigroup \(\langle A\rangle\) generated by \(A\). The genus \(g(A)\) is the number of positive integer elements that are not in \(\langle A\rangle\). The Frobenius problem is to find \(F(A)\) and \(g(A)\) for a given sequence \(A\). In this note, we study the Frobenius problem of \(A=\left(a,ba+d,b^2a+\frac{b^2-1}{b-1}d,...,b^ka+\frac{b^k-1}{b-1}d\right)\) and obtain formulas for \(F(A)\) and \(g(A)\) when \(a\geq k-1\). Our formulas simplifies further for some special cases, such as repunit, Mersenne and Thabit numerical semigroups. The idea is similar to that in [\cite{LiuXin23},arXiv:2306.03459].