On a linear diophantine problem of Frobenius: Extending the basis (Q1265677)

Let \(X_k= \{a_1,\dots, a_k\}\) be a subset of \(\mathbb{N}\) with \(\gcd (X_k)=1\). A natural number \(n\) is called dependent on \(X_k\), if there are nonnegative integers \(x_i\) such that \(n= \sum_{i=1}^k x_i a_i\), otherwise independent. The Frobenius number \(g(X_k)\) is the greatest independent integer. In this paper \(X_k\) is an arithmetic sequence with \(a_j= a+(j-1)d\), \(j=1,\dots, k\), \(k>1\), \(d>0\), \(a\geq k\), \(\gcd (a,d)=1\), and all independent numbers \(c\) satisfying \(g(X_k,c)= g(X_k)\) are given.