The Frobenius problem for Mersenne numerical semigroups

Patterns of ideals of numerical semigroups

“…Patterns can be used to explore the properties of the numerical semigroup admitting them. For example, the calculations of the formulae for the notable elements of Mersenne numerical semigroups in [10] rely on the fact that all Mersenne numerical semigroups generated by a consecutive sequence of Mersenne numbers admit the non-homogeneous pattern 2X 1 + 1. Similarly, the non-homogeneous patterns admitted by numerical semigroups associated to the existence of combinatorial configurations were used to improve the bounds on the conductor of these numerical semigroups.…”

On numerical semigroups closed with respect to the action of affine maps

“…(2) We setj (3) We set K(l + 1) := K(l) ∪ {m(l) − 1, M (l) + 1}, l := l + 1, and go to step (1). We notice that for each l we have that i∈K(l+1)…”

“…Also Mersenne numerical semigroups [1], namely numerical semigroups defined for any n ∈ N * as M (n) := 2 n+i − 1 : i ∈ N , are ϑ 2,1 -semigroups. In this case, setting c := 2 n − 1, we have that M (n) = G 2,1 (c).…”

The Frobenius problem for Thabit numerical semigroups

“…Finally, in this line, we make reference to a recent work [11] in which we study the Mersenne numerical semigroups that are of the form 2 n+i − 1 | i ∈ N where n is a positive integer.…”