Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10518_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="141" /> </InlineMediaObject> <EquationSource Format="TEX">$A=(a_1, a_2, \ldots , a_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a sequence of relative prime positive integers with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10518_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$a_i\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mi>i</mi> </msub> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. The Frobenius number F(A) is the largest integer not belonging to the numerical semigroup <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10518_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$\langle A\rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <mi>A</mi> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation> generated by A. The genus g(A) is the number of positive integer elements not in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10518_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$\langle A\rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <mi>A</mi> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation>. The Frobenius problem is to determine F(A) and g(A) for a given sequence A. In this paper, we study the Frobenius problem of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10518_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="313" /> </InlineMediaObject> <EquationSource Format="TEX">$A=\left( a,h_1a+b_1d,h_2a+b_2d,\ldots ,h_ka+b_kd\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>=</mo> <mfenced close=")" open="("> <mi>a</mi> <mo>,</mo> <msub> <mi>h</mi> <mn>1</mn> </msub> <mi>a</mi> <mo>+</mo> <msub> <mi>b</mi> <mn>1</mn> </msub> <mi>d</mi> <mo>,</mo> <msub> <mi>h</mi> <mn>2</mn> </msub> <mi>a</mi> <mo>+</mo> <msub> <mi>b</mi> <mn>2</mn> </msub> <mi>d</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>h</mi> <mi>k</mi> </msub> <mi>a</mi> <mo>+</mo> <msub> <mi>b</mi> <mi>k</mi> </msub> <mi>d</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> with some restrictions. An innovation is that d can be a negative integer. In particular, when <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10518_Article_IEq6.gif" Format="GIF" Height="33" Rendition="HTML" Resolution="72" Type="Linedraw" Width="323" /> </InlineMediaObject> <EquationSource Format="TEX">$A=\left( a,ba+d,b^2a+\frac{b^2-1}{b-1}d,\ldots ,b^ka+\frac{b^k-1}{b-1}d\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>=</mo> <mfenced close=")" open="("> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mi>a</mi> <mo>+</mo> <mi>d</mi> <mo>,</mo> <msup> <mi>b</mi> <mn>2</mn> </msup> <mi>a</mi> <mo>+</mo> <mfrac> <mrow> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <mi>b</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> <mi>d</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>b</mi> <mi>k</mi> </msup> <mi>a</mi> <mo>+</mo> <mfrac> <mrow> <msup> <mi>b</mi> <mi>k</mi> </msup> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <mi>b</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> <mi>d</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, we obtain formulas for F(A) and g(A) when <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10518_Article_IEq7.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="115" /> </InlineMediaObject> <EquationSource Format="TEX">$a\ge k-1-\frac{d-1}{b-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>≥</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>-</mo> <mfrac> <mrow> <mi>d</mi> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <mi>b</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. Our formulas simplify further for some special cases, such as Mersenne, Thabit, and repunit numerical semigroups. Finally, we partially solve an open problem for Proth numerical semigroups.

On the Frobenius number and genus of a collection of semigroups generalizing repunit numerical semigroups

期刊论文
发表日期：

2025年4月7日

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Feihu Liu,
Guoce Xin,
Suting Ye,
Jingjing Yin

摘要

Let $A=(a_1, a_2, \ldots , a_n)$ $A = (a_{1}, a_{2}, \dots, a_{n})$ be a sequence of relative prime positive integers with $a_i\ge 2$ $a_{i} \geq 2$ . The Frobenius number F(A) is the largest integer not belonging to the numerical semigroup $\langle A\rangle $ $⟨ A ⟩$ generated by A. The genus g(A) is the number of positive integer elements not in $\langle A\rangle $ $⟨ A ⟩$ . The Frobenius problem is to determine F(A) and g(A) for a given sequence A. In this paper, we study the Frobenius problem of $A=\left( a,h_1a+b_1d,h_2a+b_2d,\ldots ,h_ka+b_kd\right) $ $A = (a, h_{1} a + b_{1} d, h_{2} a + b_{2} d, \dots, h_{k} a + b_{k} d)$ with some restrictions. An innovation is that d can be a negative integer. In particular, when $A=\left( a,ba+d,b^2a+\frac{b^2-1}{b-1}d,\ldots ,b^ka+\frac{b^k-1}{b-1}d\right) $ $A = (a, b a + d, b^{2} a + \frac{b^{2} - 1}{b - 1} d, \dots, b^{k} a + \frac{b^{k} - 1}{b - 1} d)$ , we obtain formulas for F(A) and g(A) when $a\ge k-1-\frac{d-1}{b-1}$ $a \geq k - 1 - \frac{d - 1}{b - 1}$ . Our formulas simplify further for some special cases, such as Mersenne, Thabit, and repunit numerical semigroups. Finally, we partially solve an open problem for Proth numerical semigroups.