The p-Frobenius Problems for the Sequence of Generalized Repunits
Abstract For a non-negative integer p, the p-Frobenius number, which is one of the generalized Frobenius numbers in terms of the number of representations, is the largest integer represented in at most p ways by a linear combination of nonnegative integers of given positive integers %$a_1,a_2,\ldots...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Komatsu, Takao [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2023 |
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| Schlagwörter: |
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| Anmerkung: |
© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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| Übergeordnetes Werk: |
Enthalten in: Results in mathematics - Berlin : Springer, 1978, 79(2023), 1 vom: 27. Nov. |
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| Übergeordnetes Werk: |
volume:79 ; year:2023 ; number:1 ; day:27 ; month:11 |
| Links: |
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| DOI / URN: |
10.1007/s00025-023-02055-6 |
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| Katalog-ID: |
SPR053868471 |
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| 520 | |a Abstract For a non-negative integer p, the p-Frobenius number, which is one of the generalized Frobenius numbers in terms of the number of representations, is the largest integer represented in at most p ways by a linear combination of nonnegative integers of given positive integers %$a_1,a_2,\ldots ,a_k%$ with %$\gcd (a_1,a_2,\ldots ,a_k)=1%$. When %$p=0%$, it reduces to the classical Frobenius number. One of the most natural questions is to find a closed explicit form of the Frobenius number. When %$k=2%$, its explicit formula was discovered in the nineteenth century. When %$k\ge 3%$, explicit formulas are very difficult to obtain even for %$p=0%$. The case of %$p>0%$ is even more difficult, and until recently there was no explicit formula for the Frobenius number even in a single case. However, we finally found explicit formulas for the p-Frobenius number, such as in the case of repunit. In this paper, we give closed formulas for the p-Frobenius number for the generalized repunit. The method is to analyze the structure of the p-Apéry set, which is a more general Apéry set. In the generalized repunit, the structure of the Apéry set is similar, but the position of the element that takes the maximum value is different, making it more difficult to identify. | ||
| 650 | 4 | |a -Frobenius numbers |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a denumerants |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a generalized repunits |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Apéry sets |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a numerical semigroup |7 (dpeaa)DE-He213 | |
| 700 | 1 | |a Laohakosol, Vichian |4 aut | |
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10.1007/s00025-023-02055-6 doi (DE-627)SPR053868471 (SPR)s00025-023-02055-6-e DE-627 ger DE-627 rakwb eng Komatsu, Takao verfasserin (orcid)0000-0001-6204-5368 aut The p-Frobenius Problems for the Sequence of Generalized Repunits 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract For a non-negative integer p, the p-Frobenius number, which is one of the generalized Frobenius numbers in terms of the number of representations, is the largest integer represented in at most p ways by a linear combination of nonnegative integers of given positive integers %$a_1,a_2,\ldots ,a_k%$ with %$\gcd (a_1,a_2,\ldots ,a_k)=1%$. When %$p=0%$, it reduces to the classical Frobenius number. One of the most natural questions is to find a closed explicit form of the Frobenius number. When %$k=2%$, its explicit formula was discovered in the nineteenth century. When %$k\ge 3%$, explicit formulas are very difficult to obtain even for %$p=0%$. The case of %$p>0%$ is even more difficult, and until recently there was no explicit formula for the Frobenius number even in a single case. However, we finally found explicit formulas for the p-Frobenius number, such as in the case of repunit. In this paper, we give closed formulas for the p-Frobenius number for the generalized repunit. The method is to analyze the structure of the p-Apéry set, which is a more general Apéry set. In the generalized repunit, the structure of the Apéry set is similar, but the position of the element that takes the maximum value is different, making it more difficult to identify. -Frobenius numbers (dpeaa)DE-He213 denumerants (dpeaa)DE-He213 generalized repunits (dpeaa)DE-He213 Apéry sets (dpeaa)DE-He213 numerical semigroup (dpeaa)DE-He213 Laohakosol, Vichian aut Enthalten in Results in mathematics Berlin : Springer, 1978 79(2023), 1 vom: 27. Nov. (DE-627)327046066 (DE-600)2043519-8 1420-9012 nnns volume:79 year:2023 number:1 day:27 month:11 https://dx.doi.org/10.1007/s00025-023-02055-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 79 2023 1 27 11 |
| spelling |
10.1007/s00025-023-02055-6 doi (DE-627)SPR053868471 (SPR)s00025-023-02055-6-e DE-627 ger DE-627 rakwb eng Komatsu, Takao verfasserin (orcid)0000-0001-6204-5368 aut The p-Frobenius Problems for the Sequence of Generalized Repunits 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract For a non-negative integer p, the p-Frobenius number, which is one of the generalized Frobenius numbers in terms of the number of representations, is the largest integer represented in at most p ways by a linear combination of nonnegative integers of given positive integers %$a_1,a_2,\ldots ,a_k%$ with %$\gcd (a_1,a_2,\ldots ,a_k)=1%$. When %$p=0%$, it reduces to the classical Frobenius number. One of the most natural questions is to find a closed explicit form of the Frobenius number. When %$k=2%$, its explicit formula was discovered in the nineteenth century. When %$k\ge 3%$, explicit formulas are very difficult to obtain even for %$p=0%$. The case of %$p>0%$ is even more difficult, and until recently there was no explicit formula for the Frobenius number even in a single case. However, we finally found explicit formulas for the p-Frobenius number, such as in the case of repunit. In this paper, we give closed formulas for the p-Frobenius number for the generalized repunit. The method is to analyze the structure of the p-Apéry set, which is a more general Apéry set. In the generalized repunit, the structure of the Apéry set is similar, but the position of the element that takes the maximum value is different, making it more difficult to identify. -Frobenius numbers (dpeaa)DE-He213 denumerants (dpeaa)DE-He213 generalized repunits (dpeaa)DE-He213 Apéry sets (dpeaa)DE-He213 numerical semigroup (dpeaa)DE-He213 Laohakosol, Vichian aut Enthalten in Results in mathematics Berlin : Springer, 1978 79(2023), 1 vom: 27. Nov. (DE-627)327046066 (DE-600)2043519-8 1420-9012 nnns volume:79 year:2023 number:1 day:27 month:11 https://dx.doi.org/10.1007/s00025-023-02055-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 79 2023 1 27 11 |
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10.1007/s00025-023-02055-6 doi (DE-627)SPR053868471 (SPR)s00025-023-02055-6-e DE-627 ger DE-627 rakwb eng Komatsu, Takao verfasserin (orcid)0000-0001-6204-5368 aut The p-Frobenius Problems for the Sequence of Generalized Repunits 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract For a non-negative integer p, the p-Frobenius number, which is one of the generalized Frobenius numbers in terms of the number of representations, is the largest integer represented in at most p ways by a linear combination of nonnegative integers of given positive integers %$a_1,a_2,\ldots ,a_k%$ with %$\gcd (a_1,a_2,\ldots ,a_k)=1%$. When %$p=0%$, it reduces to the classical Frobenius number. One of the most natural questions is to find a closed explicit form of the Frobenius number. When %$k=2%$, its explicit formula was discovered in the nineteenth century. When %$k\ge 3%$, explicit formulas are very difficult to obtain even for %$p=0%$. The case of %$p>0%$ is even more difficult, and until recently there was no explicit formula for the Frobenius number even in a single case. However, we finally found explicit formulas for the p-Frobenius number, such as in the case of repunit. In this paper, we give closed formulas for the p-Frobenius number for the generalized repunit. The method is to analyze the structure of the p-Apéry set, which is a more general Apéry set. In the generalized repunit, the structure of the Apéry set is similar, but the position of the element that takes the maximum value is different, making it more difficult to identify. -Frobenius numbers (dpeaa)DE-He213 denumerants (dpeaa)DE-He213 generalized repunits (dpeaa)DE-He213 Apéry sets (dpeaa)DE-He213 numerical semigroup (dpeaa)DE-He213 Laohakosol, Vichian aut Enthalten in Results in mathematics Berlin : Springer, 1978 79(2023), 1 vom: 27. Nov. (DE-627)327046066 (DE-600)2043519-8 1420-9012 nnns volume:79 year:2023 number:1 day:27 month:11 https://dx.doi.org/10.1007/s00025-023-02055-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 79 2023 1 27 11 |
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10.1007/s00025-023-02055-6 doi (DE-627)SPR053868471 (SPR)s00025-023-02055-6-e DE-627 ger DE-627 rakwb eng Komatsu, Takao verfasserin (orcid)0000-0001-6204-5368 aut The p-Frobenius Problems for the Sequence of Generalized Repunits 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract For a non-negative integer p, the p-Frobenius number, which is one of the generalized Frobenius numbers in terms of the number of representations, is the largest integer represented in at most p ways by a linear combination of nonnegative integers of given positive integers %$a_1,a_2,\ldots ,a_k%$ with %$\gcd (a_1,a_2,\ldots ,a_k)=1%$. When %$p=0%$, it reduces to the classical Frobenius number. One of the most natural questions is to find a closed explicit form of the Frobenius number. When %$k=2%$, its explicit formula was discovered in the nineteenth century. When %$k\ge 3%$, explicit formulas are very difficult to obtain even for %$p=0%$. The case of %$p>0%$ is even more difficult, and until recently there was no explicit formula for the Frobenius number even in a single case. However, we finally found explicit formulas for the p-Frobenius number, such as in the case of repunit. In this paper, we give closed formulas for the p-Frobenius number for the generalized repunit. The method is to analyze the structure of the p-Apéry set, which is a more general Apéry set. In the generalized repunit, the structure of the Apéry set is similar, but the position of the element that takes the maximum value is different, making it more difficult to identify. -Frobenius numbers (dpeaa)DE-He213 denumerants (dpeaa)DE-He213 generalized repunits (dpeaa)DE-He213 Apéry sets (dpeaa)DE-He213 numerical semigroup (dpeaa)DE-He213 Laohakosol, Vichian aut Enthalten in Results in mathematics Berlin : Springer, 1978 79(2023), 1 vom: 27. Nov. (DE-627)327046066 (DE-600)2043519-8 1420-9012 nnns volume:79 year:2023 number:1 day:27 month:11 https://dx.doi.org/10.1007/s00025-023-02055-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 79 2023 1 27 11 |
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10.1007/s00025-023-02055-6 doi (DE-627)SPR053868471 (SPR)s00025-023-02055-6-e DE-627 ger DE-627 rakwb eng Komatsu, Takao verfasserin (orcid)0000-0001-6204-5368 aut The p-Frobenius Problems for the Sequence of Generalized Repunits 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract For a non-negative integer p, the p-Frobenius number, which is one of the generalized Frobenius numbers in terms of the number of representations, is the largest integer represented in at most p ways by a linear combination of nonnegative integers of given positive integers %$a_1,a_2,\ldots ,a_k%$ with %$\gcd (a_1,a_2,\ldots ,a_k)=1%$. When %$p=0%$, it reduces to the classical Frobenius number. One of the most natural questions is to find a closed explicit form of the Frobenius number. When %$k=2%$, its explicit formula was discovered in the nineteenth century. When %$k\ge 3%$, explicit formulas are very difficult to obtain even for %$p=0%$. The case of %$p>0%$ is even more difficult, and until recently there was no explicit formula for the Frobenius number even in a single case. However, we finally found explicit formulas for the p-Frobenius number, such as in the case of repunit. In this paper, we give closed formulas for the p-Frobenius number for the generalized repunit. The method is to analyze the structure of the p-Apéry set, which is a more general Apéry set. In the generalized repunit, the structure of the Apéry set is similar, but the position of the element that takes the maximum value is different, making it more difficult to identify. -Frobenius numbers (dpeaa)DE-He213 denumerants (dpeaa)DE-He213 generalized repunits (dpeaa)DE-He213 Apéry sets (dpeaa)DE-He213 numerical semigroup (dpeaa)DE-He213 Laohakosol, Vichian aut Enthalten in Results in mathematics Berlin : Springer, 1978 79(2023), 1 vom: 27. Nov. (DE-627)327046066 (DE-600)2043519-8 1420-9012 nnns volume:79 year:2023 number:1 day:27 month:11 https://dx.doi.org/10.1007/s00025-023-02055-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 79 2023 1 27 11 |
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Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract For a non-negative integer p, the p-Frobenius number, which is one of the generalized Frobenius numbers in terms of the number of representations, is the largest integer represented in at most p ways by a linear combination of nonnegative integers of given positive integers %$a_1,a_2,\ldots ,a_k%$ with %$\gcd (a_1,a_2,\ldots ,a_k)=1%$. When %$p=0%$, it reduces to the classical Frobenius number. One of the most natural questions is to find a closed explicit form of the Frobenius number. When %$k=2%$, its explicit formula was discovered in the nineteenth century. 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Komatsu, Takao |
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The p-Frobenius Problems for the Sequence of Generalized Repunits -Frobenius numbers (dpeaa)DE-He213 denumerants (dpeaa)DE-He213 generalized repunits (dpeaa)DE-He213 Apéry sets (dpeaa)DE-He213 numerical semigroup (dpeaa)DE-He213 |
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p-frobenius problems for the sequence of generalized repunits |
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The p-Frobenius Problems for the Sequence of Generalized Repunits |
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Abstract For a non-negative integer p, the p-Frobenius number, which is one of the generalized Frobenius numbers in terms of the number of representations, is the largest integer represented in at most p ways by a linear combination of nonnegative integers of given positive integers %$a_1,a_2,\ldots ,a_k%$ with %$\gcd (a_1,a_2,\ldots ,a_k)=1%$. When %$p=0%$, it reduces to the classical Frobenius number. One of the most natural questions is to find a closed explicit form of the Frobenius number. When %$k=2%$, its explicit formula was discovered in the nineteenth century. When %$k\ge 3%$, explicit formulas are very difficult to obtain even for %$p=0%$. The case of %$p>0%$ is even more difficult, and until recently there was no explicit formula for the Frobenius number even in a single case. However, we finally found explicit formulas for the p-Frobenius number, such as in the case of repunit. In this paper, we give closed formulas for the p-Frobenius number for the generalized repunit. The method is to analyze the structure of the p-Apéry set, which is a more general Apéry set. In the generalized repunit, the structure of the Apéry set is similar, but the position of the element that takes the maximum value is different, making it more difficult to identify. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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Abstract For a non-negative integer p, the p-Frobenius number, which is one of the generalized Frobenius numbers in terms of the number of representations, is the largest integer represented in at most p ways by a linear combination of nonnegative integers of given positive integers %$a_1,a_2,\ldots ,a_k%$ with %$\gcd (a_1,a_2,\ldots ,a_k)=1%$. When %$p=0%$, it reduces to the classical Frobenius number. One of the most natural questions is to find a closed explicit form of the Frobenius number. When %$k=2%$, its explicit formula was discovered in the nineteenth century. When %$k\ge 3%$, explicit formulas are very difficult to obtain even for %$p=0%$. The case of %$p>0%$ is even more difficult, and until recently there was no explicit formula for the Frobenius number even in a single case. However, we finally found explicit formulas for the p-Frobenius number, such as in the case of repunit. In this paper, we give closed formulas for the p-Frobenius number for the generalized repunit. The method is to analyze the structure of the p-Apéry set, which is a more general Apéry set. In the generalized repunit, the structure of the Apéry set is similar, but the position of the element that takes the maximum value is different, making it more difficult to identify. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
| abstract_unstemmed |
Abstract For a non-negative integer p, the p-Frobenius number, which is one of the generalized Frobenius numbers in terms of the number of representations, is the largest integer represented in at most p ways by a linear combination of nonnegative integers of given positive integers %$a_1,a_2,\ldots ,a_k%$ with %$\gcd (a_1,a_2,\ldots ,a_k)=1%$. When %$p=0%$, it reduces to the classical Frobenius number. One of the most natural questions is to find a closed explicit form of the Frobenius number. When %$k=2%$, its explicit formula was discovered in the nineteenth century. When %$k\ge 3%$, explicit formulas are very difficult to obtain even for %$p=0%$. The case of %$p>0%$ is even more difficult, and until recently there was no explicit formula for the Frobenius number even in a single case. However, we finally found explicit formulas for the p-Frobenius number, such as in the case of repunit. In this paper, we give closed formulas for the p-Frobenius number for the generalized repunit. The method is to analyze the structure of the p-Apéry set, which is a more general Apéry set. In the generalized repunit, the structure of the Apéry set is similar, but the position of the element that takes the maximum value is different, making it more difficult to identify. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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The p-Frobenius Problems for the Sequence of Generalized Repunits |
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| score |
7.402421 |