Some problems concerning the Frobenius number for extensions of an arithmetic progression

Abstract

For positive and relative prime set of integers $A=\{a_1,\ldots ,a_k\}$, let ${\varGamma }(A)$ denote the set of integers of the form $a_1x_1+\cdots +a_kx_k$ with each $x_i \ge 0$. It is well known that ${\varGamma }^c(A)=\mathbb {N}\setminus {\varGamma }(A)$ is a finite set, so that ${\texttt {g}}(A)$, which denotes the largest integer in ${\varGamma }^c(A)$, is well defined. Let $A=AP(a,d,k)$ denote the set $\{a,a+d,\ldots ,a+(k-1)d\}$ of integers in arithmetic progression, and let $\gcd (a,d)=1$. We (i) determine the set $A^+=\left\{ b \in {\varGamma }^c(A): {\texttt {g}}(A \cup \{b\})={\texttt {g}}(A) \right\}$; (ii) determine a subset $\overline{A^+}$ of ${\varGamma }^c(A)$ of largest cardinality such that $A \cup \overline{A^+}$ is an independent set and ${\texttt {g}}(A\,\cup \,\overline{A^+})={\texttt {g}}(A)$; and (iii) determine ${\texttt {g}}(A \cup \{b\})$ for some class of values of b that includes results of some recent work.