On Pythagorean triplet semigroups

In this note we explain the two pseudo-Frobenius numbers for \langle m^2-n^2,m^2+n^2,2mn\rangle where m and n are two coprime numbers of different parity. This lets us give an Apéry set for these numerical semigroups.

Compositions of a numerical semigroup

Given a numerical semigroup S, a nonnegative integer a and m ∈ S ∖ {0}, we introduce the set C(S, a, m) = {s + aw(s mod m) | s ∈ S}, where {w(0), w(1), ⋯, w(m – 1)} is the Apéry set of m in S. In this paper we characterize the pairs (a, m) such that C(S, a, m) is a numerical semigroup. We study the principal invariants of C(S, a, m) which are given explicitly in terms of invariants of S. We also characterize the compositions C(S, a, m) that are symmetric, pseudo-symmetric and almost symmetric. Finally, a result about compliance to Wilf’s conjecture of C(S, a, m) is given.