Counting numerical semigroups by genus and even gaps

Let n(g) be the number of numerical semigroups of genus g. We present an approach to compute n(g) by using even gaps, and the question: Is it true that n(g+1) > n(g) ? is investigated. Let N-gamma(g) be the number of numerical semigroups of genus g whose number of even gaps equals gamma. We show that N-gamma(g) = N-gamma (3 gamma) for gamma <= left perpendicularg/3right perpendicular and N-gamma(g) = 0 for gamma > left perpendicular2g/3right perpendicular; thus the question above is true provided that N-gamma(g + 1) > N-gamma(g) for y = left perpendicularg/3right perpendicular + 1, ... , left perpendicular2g/3right perpendicular. We also show that N-gamma (3 gamma) coincides with f(gamma), the number introduced by Bras-Amords (2012) in connection with semigroup-closed sets. Finally, the stronger possibility f(gamma) similar to phi(2 gamma) arises being phi = (1 + root 5)/2 the golden number. (C) 2017 Elsevier B.V. All rights reserved.