On the Frobenius number of certain numerical semigroups

Abstract

Let $0<\lambda \le 1$, $\lambda \notin \left\{\frac{2}{4},\frac{2}{7},\frac{2}{10},\frac{2}{13},\dots \right\}$, be a real and $p$ a prime number, with $[p,p+\lambda p]$ containing at least two primes. Denote by $f_{\lambda }(p)$ the largest integer which cannot be written as a sum of primes from $[p,p+\lambda p]$. Then \[f_λ(p)\sim\left\lfloor2+\frac2λ\right\rfloor\cdot p\text{, as }p\text{ goes to infinity.}\] Further a question of Wilf about the 'Money-Changing Problem' has a positive answer for all semigroups of multiplicity $p$ containing the primes from $[p,2p]$. In particular, this holds for the semigroup generated by all primes not less than $p$. The latter special case was already shown in a previous paper.

15 pages, 2 figures

References in corpus (4)

• Array Programming with NumPy
• Wilf's conjecture and Macaulay's theorem
• Vinogradov's theorem with almost equal summands
• Numerical Semigroups generated by Primes