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.