The Square Frobenius Number

Let $S=⟨s_1,\dots ,s_n⟩$ be a numerical semigroup generated by the relatively prime positive integers $s_1,\dots ,s_n$. Let $k⩾2$ be an integer. In this paper, we consider the following $k$-power variant of the Frobenius number of $S$ defined as $${}^{k\text{\hspace{0.17em}⁣}}r\text{\hspace{0.17em}⁣}\left(S\right):=\text{ the largest }k\text{-power integer not belonging to }S\mathrm{.}$$In this paper, we investigate the case $k=2$. We give an upper bound for ${}^{2\text{\hspace{0.17em}⁣}}r\text{\hspace{0.17em}⁣}\left(S_A\right)$ for an infinite family of semigroups $S_A$ generated by {\em 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 ${}^{2\text{\hspace{0.17em}⁣}}r\text{\hspace{0.17em}⁣}\left(⟨s_1,s_1+d⟩\right)$ 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.