On the Frobenius Number and Genus of a Collection of Semigroups Generalizing Repunit Numerical Semigroups

Let $A=(a_1,a_2,\dots ,a_n)$ be a sequence of relative prime positive integers with $a_i\ge 2$. The Frobenius number $F(A)$ is the largest integer not belonging to the numerical semigroup $⟨A⟩$ generated by $A$. The genus $g(A)$ is the number of positive integer elements not in $⟨A⟩$. The Frobenius problem is to determine $F(A)$ and $g(A)$ for a given sequence $A$. In this paper, we study the Frobenius problem of $A=(a,h_{1}a+b_{1}d,h_{2}a+b_{2}d,\dots ,h_{k}a+b_{k}d)$ with some restrictions. An innovation is that $d$ can be a negative integer. In particular, when $A=(a,ba+d,b^{2}a+\frac{b^2-1}{b-1}d,\dots ,b^{k}a+\frac{b^k-1}{b-1}d)$, we obtain formulas for $F(A)$ and $g(A)$ when $a\ge k-1-\frac{d-1}{b-1}$. Our formulas simplify further for some special cases, such as Mersenne, Thabit, and repunit numerical semigroups. Finally, we partially solve an open problem for the Proth numerical semigroup.