The square Frobenius number

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