The square Frobenius number

Abstract

Let \(S=\left\langle s_1,\ldots ,s_n\right\rangle \) be a numerical semigroup generated by the relatively prime positive integers \(s_1,\ldots ,s_n\). Let \(k\geqslant 2\) be an integer. In this paper, we consider the following k-power variant of the Frobenius number of S defined as $$\begin{aligned} {}^{k\!}r\!\left( S\right) := \text { the largest } k \text {-power integer not belonging to } S. \end{aligned}$$In this paper, we investigate the case \(k=2\). We give an upper bound for \({}^{2\!}r\!\left( S_A\right) \) for an infinite family of semigroups \(S_A\) generated by arithmetic progressions. The latter turns out to be the exact value of \({}^{2\!}r\!\left( \left\langle s_1,s_2\right\rangle \right) \) under certain conditions. We present an exact formula for \({}^{2\!}r\!\left( \left\langle s_1,s_1+d \right\rangle \right) \) when \(d=3,4\) and 5, study \({}^{2\!}r\!\left( \left\langle s_1,s_1+1 \right\rangle \right) \) and \({}^{2\!}r\!\left( \left\langle s_1,s_1+2 \right\rangle \right) \) and put forward two relevant conjectures. We finally discuss some related questions.