  In this paper, we introduce and study the numerical semigroups generated by
$\{a_1, a_2, \ldots \} \subset \mathbb{N}$ 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 $i \geq 2$, 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 Ap\'ery sets, the genus or the
type.
