In this paper, we introduce and study the numerical semigroups generated by {a 1 , a 2 , . . .} ⊂ N such that a 1 is the repunit number in base b > 1 of length n > 1 and a i − a i−1 = a b i−2 , for every i ≥ 2, where a is a positive integer relatively prime with a 1 . These numerical semigroups generalize the repunit numerical semigroups among many others. We show that they have interesting properties such as being homogeneous and Wilf. Moreover, we solve the Frobenius problem for this family, by giving a closed formula for the Frobenius number in terms of a, b and n, and compute other usual invariants such as the Apéry sets, the genus or the type.

In this paper, we introduce and study the numerical semigroups generated by $$\{a_1, a_2, \ldots \} \subset {\mathbb {N}}$$ { a 1 , a 2 , … } ⊂ N such that $$a_1$$ a 1 is the repunit number in base $$b > 1$$ b > 1 of length $$n > 1$$ n > 1 and $$a_i - a_{i-1} = a\, b^{i-2},$$ a i - a i - 1 = a b i - 2 , for every $$i \ge 2$$ i ≥ 2 , where a is a positive integer relatively prime with $$a_1$$ a 1 . These numerical semigroups generalize the repunit numerical semigroups among many others. We show that they have interesting properties such as being homogeneous and Wilf. Moreover, we solve the Frobenius problem for this family, by giving a closed formula for the Frobenius number in terms of a, b and n, and compute other usual invariants such as the Apéry sets, the genus or the type.

Let a,b and n>1 be three positive integers such that a and ∑j=0n−1bj are relatively prime. In this paper, we prove that the toric ideal I associated to the submonoid of N generated by {∑j=0n−1bj}∪{∑j=0n−1bj+a∑j=0i−2bj∣i=2,…,n} is determinantal. Moreover, we prove that for n>3, the ideal I has a unique minimal system of generators if and only if a<b−1.

Let $a, b$ and $n &gt; 1$ be three positive integers such that $a$ and $\sum_{j=0}^{n-1} b^j$ are relatively prime. In this paper, we prove that the toric ideal $I$ associated to the submonoid of $\mathbb{N}$ generated by $\{\sum_{j=0}^{n-1} b^j\} \cup \{\sum_{j=0}^{n-1} b^j + a\, \sum_{j=0}^{i-2} b^j \mid i = 2, \ldots, n\}$ is determinantal. Moreover, we prove that for $n &gt; 3$, the ideal $I$ has a unique minimal system of generators if and only if $a &lt; b-1$.