The Frobenius number in the set of numerical semigroups with fixed multiplicity and genus

Electronic version of an article published as International Journal of Number Theory, 2017, Vol. 13, No. 04 : pp. 1003-1011 https://doi.org/10.1142/S1793042117500531 © copyright World Scientific Publishing Company http://www.worldscientific.com/We compute all possible numbers that are the Frobenius number of a numerical semigroup when multiplicity and genus are fixed. Moreover, we construct explicitly numerical semigroups in each case.Both authors are supported by the project MTM2014-55367-P, which is funded by Ministerio de Economía y Competitividad and Fondo Europeo de Desarrollo Regional FEDER, and by the Junta de Andalucía Grant Number FQM-343. The second author is also partially supported by Junta de Andalucía/Feder Grant Number FQM-5849

Progression bases for finite cyclic groups

A set A of integers is an additive basis modulo n if every integer is congruent mod n to a sum of at most h elements of A, repetitions being allowed. The set A is a basis of order h in case h is minimal. In this paper we study the order of bases of the form {a,2a,...,ka} ∪ {b,2b,...,lb} and of the form {a,a + b,a + 2b,...,a + kb}, where a,b are integers satisfying gcd(a,b,n) = 1.