Revisiting Wilf’s Question for Numerical Semigroups S3 and Inequalities for Rescaled Genera

We consider numerical semigroups S3=⟨d1,d2,d3⟩, which are minimally generated by three positive integers. We revisit the Wilf question for S3 and, making use of identities for degrees of syzygies of such semigroups, we provide a short proof of existence of an affirmative answer. Finally, we find the upper and lower bounds for the rescaled genera of numerical semigroups S3.