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 $s_1,\ldots,s_n$. 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 $${}^{k\!}r\!\left(S\right):= \text{ the largest } k \text{-power integer not belonging to } S.$$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 {\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 ${}^{2\!}r\!\left(\left\langle s_1,s_1+d \right\rangle\right)$ 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.