On the number of numerical semigroups 〈a,b〉 of prime power genus - Eliahou, Shalom; Ramírez Alfonsín, Jorge (Semigroup forum, 2013)
Given g ≥1, the number n ( g ) of numerical semigroups S ⊂ℕ of genus |ℕ∖ S | equal to g is the subject of challenging conjectures of Bras-Amorós. In this paper, we focus on the counting function n ( g ,2) of two-generator numerical semigroups of genus g , which is known to also count certain special factorizations of 2 g . Further focusing on the case g = p k for any odd prime p and k ≥1, we show that n ( p k ,2) only depends on the class of p modulo a certain explicit modulus M ( k ). The main ingredient is a reduction of to a simpler form, using the continued fraction of α / β . We treat the case k =9 in detail and show explicitly how n ( p 9 ,2) depends on the class of .