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Math."],"published-print":{"date-parts":[[2023,2]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper, we introduce and study the numerical semigroups generated by <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\{a_1, a_2, \\ldots \\} \\subset {\\mathbb {N}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>{<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>a<\/mml:mi>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>a<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mo>\u2026<\/mml:mo>\n                    <mml:mo>}<\/mml:mo>\n                    <mml:mo>\u2282<\/mml:mo>\n                    <mml:mi>N<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> such that <jats:inline-formula><jats:alternatives><jats:tex-math>$$a_1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>a<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is the repunit number in base <jats:inline-formula><jats:alternatives><jats:tex-math>$$b &gt; 1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>b<\/mml:mi>\n                    <mml:mo>&gt;<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of length <jats:inline-formula><jats:alternatives><jats:tex-math>$$n &gt; 1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>&gt;<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$a_i - a_{i-1} = a\\, b^{i-2},$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>a<\/mml:mi>\n                      <mml:mi>i<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>a<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mi>i<\/mml:mi>\n                        <mml:mo>-<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msub>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mi>a<\/mml:mi>\n                    <mml:mspace\/>\n                    <mml:msup>\n                      <mml:mi>b<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mi>i<\/mml:mi>\n                        <mml:mo>-<\/mml:mo>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                    <mml:mo>,<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for every <jats:inline-formula><jats:alternatives><jats:tex-math>$$i \\ge 2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>i<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where <jats:italic>a<\/jats:italic> is a positive integer relatively prime with <jats:inline-formula><jats:alternatives><jats:tex-math>$$a_1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>a<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. These numerical semigroups generalize the repunit numerical semigroups among many others. We show that they have interesting properties such as being homogeneous and Wilf. Moreover, we solve the Frobenius problem for this family, by giving a closed formula for the Frobenius number in terms of <jats:italic>a<\/jats:italic>,\u00a0<jats:italic>b<\/jats:italic> and <jats:italic>n<\/jats:italic>, and compute other usual invariants such as the Ap\u00e9ry sets, the genus or the type.<\/jats:p>","DOI":"10.1007\/s00009-022-02233-w","type":"journal-article","created":{"date-parts":[[2022,12,3]],"date-time":"2022-12-03T20:02:26Z","timestamp":1670097746000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["The Frobenius Problem for Generalized Repunit Numerical Semigroups"],"prefix":"10.1007","volume":"20","author":[{"given":"Manuel B.","family":"Branco","sequence":"first","affiliation":[]},{"given":"Isabel","family":"Cola\u00e7o","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3173-5934","authenticated-orcid":false,"given":"Ignacio","family":"Ojeda","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,12,3]]},"reference":[{"issue":"24","key":"2233_CR1","doi-asserted-by":"publisher","first-page":"3204","DOI":"10.3390\/math9243204","volume":"9","author":"MB Branco","year":"2021","unstructured":"Branco, M.B., Cola\u00e7o, I., Ojeda, I.: Minimal binomial systems of generators for the ideals of certain monomial curves. 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