The Frobenius problem for classes of polynomial solvability. (Q1810026)

Let \(A=\{a_1,\dots,a_n\}\) be positive integers with \(a_i\geq 2\) and such that their greatest common divisor is one. An integer \(s\) is representable as a nonnegative integral combination of \(a_1,\dots,a_n\) if there exist integers \(x_i\geq 0\) such that \(s=\sum^n_{i=1}x_ia_i\). The celebrated Frobenius problem is to find the largest natural number that is not representable as a nonnegative integral combination of \(a_1,\dots, a_n\). We denote by \(g(A)\) this number. In this paper, the author estimates \(g(A)\) when \(A\) is an almost chain sequence yielding to some formulas for particular sequences, for instance, for \(g(a,a+1,a+2, a+b, a+2b)\) where integers \(b<a\) and \(a+1\geq(1+b) \lfloor-\frac ab\rfloor\).