On the extremal aspect of the Frobenius problem (Q1906122)

Concerning the Frobenius number \(g= g(a_1, \dots, a_n)\) (cf. the review above) \textit{P. Erdös} and \textit{R. L. Graham} [Acta Arith. 21, 399-408 (1972; Zbl 0246.10010)] defined \(g(n, \ell)= \max_{a_n= \ell} g\), the maximum being taken over all sets of \(n\) coefficients with the maximal coefficient \(\ell\) \((\ell \geq n)\). It is known that \(g(n, \ell)= 2(\ell- n)-1\) if \(\ell \leq 2n- 1\), and \(g(n, 2n)= 2n +1\). In this paper it is shown for \(2n< \ell\leq 3n-2\) \[ g(n, \ell)= \begin{cases} 2(2\ell- 3n)+1 &\quad \text{if } \ell\not\equiv 2\bmod 3\\ 2(2\ell- 3n)-1 &\quad \text{if } \ell\equiv 2 \bmod 3.\end{cases} \] {}.