Let S=s1,…,sn\documentclass[12pt]{minimal}
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\begin{document}$$S=\left\langle s_1,\ldots ,s_n\right\rangle $$\end{document} be a numerical semigroup generated by the relatively prime positive integers s1,…,sn\documentclass[12pt]{minimal}
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\begin{document}$$s_1,\ldots ,s_n$$\end{document}. Let k⩾2\documentclass[12pt]{minimal}
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\begin{document}$$k\geqslant 2$$\end{document} be an integer. In this paper, we consider the following k-power variant of the Frobenius number of S defined as krS:=the largestk-power integer not belonging toS.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {}^{k\!}r\!\left( S\right) := \text { the largest } k \text {-power integer not belonging to } S. \end{aligned}$$\end{document}In this paper, we investigate the case k=2\documentclass[12pt]{minimal}
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\begin{document}$$k=2$$\end{document}. We give an upper bound for 2rSA\documentclass[12pt]{minimal}
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\begin{document}$${}^{2\!}r\!\left( S_A\right) $$\end{document} for an infinite family of semigroups SA\documentclass[12pt]{minimal}
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\begin{document}$$S_A$$\end{document} generated by arithmetic progressions. The latter turns out to be the exact value of 2rs1,s2\documentclass[12pt]{minimal}
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\begin{document}$${}^{2\!}r\!\left( \left\langle s_1,s_2\right\rangle \right) $$\end{document} under certain conditions. We present an exact formula for 2rs1,s1+d\documentclass[12pt]{minimal}
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\begin{document}$${}^{2\!}r\!\left( \left\langle s_1,s_1+d \right\rangle \right) $$\end{document} when d=3,4\documentclass[12pt]{minimal}
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\begin{document}$$d=3,4$$\end{document} and 5, study 2rs1,s1+1\documentclass[12pt]{minimal}
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\begin{document}$${}^{2\!}r\!\left( \left\langle s_1,s_1+1 \right\rangle \right) $$\end{document} and 2rs1,s1+2\documentclass[12pt]{minimal}
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\begin{document}$${}^{2\!}r\!\left( \left\langle s_1,s_1+2 \right\rangle \right) $$\end{document} and put forward two relevant conjectures. We finally discuss some related questions.
