The Frobenius number associated with the number of representations for sequences of repunits

Takao Komatsu

doi:10.5802/crmath.394

Combinatoire, Théorie des nombres

The Frobenius number associated with the number of representations for sequences of repunits

Takao Komatsu ¹

¹ Department of Mathematical Sciences, School of Science, Zhejiang Sci-Tech University, Hangzhou 310018 China

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 73-89.

Résumé

The generalized Frobenius number is the largest integer represented in at most $p$ ways by a linear combination of nonnegative integers of given positive integers $a_{1}, a_{2}, \dots, a_{k}$ . When $p = 0$ , it reduces to the classical Frobenius number. In this paper, we give the generalized Frobenius number when $a_{j} = (b^{n + j - 1} - 1) / (b - 1)$ ( $b \geq 2$ ) as a generalization of the result of $p = 0$ in [16].

Reçu le : 2021-11-15
Révisé le : 2022-04-20
Accepté le : 2022-06-08
Publié le : 2023-01-12

DOI : 10.5802/crmath.394

Classification : 11D07, 05A15, 05A17, 05A19, 11B68, 11D04, 11P81

Affiliations des auteurs :

Takao Komatsu ¹

¹ Department of Mathematical Sciences, School of Science, Zhejiang Sci-Tech University, Hangzhou 310018 China

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {The {Frobenius} number associated with the number of representations for sequences of repunits},
     journal = {Comptes Rendus. Math\'ematique},
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     year = {2023},
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     language = {en},
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Takao Komatsu. The Frobenius number associated with the number of representations for sequences of repunits. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 73-89. doi : 10.5802/crmath.394. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.394/

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Jiaxin Mu; Takao Komatsu p-Numerical Semigroups of Triples from the Three-Term Recurrence Relations, Axioms, Volume 13 (2024) no. 9, p. 608 | DOI:10.3390/axioms13090608
TAKAO KOMATSU; NEHA GUPTA; MANOJ UPRETI FROBENIUS NUMBERS ASSOCIATED WITH DIOPHANTINE TRIPLES OF $x^{2} + y^{2} = z^{3}$ , Bulletin of the Australian Mathematical Society (2024), p. 1 | DOI:10.1017/s0004972724000960
Takao Komatsu; Haotian Ying p-Numerical semigroups with p-symmetric properties, Journal of Algebra and Its Applications, Volume 23 (2024) no. 13 | DOI:10.1142/s0219498824502165
Takao Komatsu; Fatih Yilmaz The p-Frobenius number for the triple of certain quadratic numbers, Logic Journal of the IGPL (2024) | DOI:10.1093/jigpal/jzae125
Takao Komatsu; Ruze Yin p-Numerical Semigroups of the Triples of the Sequence $(a^{n} - b^{n}) / (a - b)$ , Mathematical Methods for Engineering Applications, Volume 439 (2024), p. 15 | DOI:10.1007/978-3-031-49218-1_2
Takao Komatsu; Vichian Laohakosol The p-Frobenius Problems for the Sequence of Generalized Repunits, Results in Mathematics, Volume 79 (2024) no. 1 | DOI:10.1007/s00025-023-02055-6
Takao Komatsu On the determination of p-Frobenius and related numbers using the p-Apéry set, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, Volume 118 (2024) no. 2 | DOI:10.1007/s13398-024-01556-5
Ruze Yin; Takao Komatsu Frobenius Numbers Associated with Diophantine Triples of x2-y2=zr, Symmetry, Volume 16 (2024) no. 7, p. 855 | DOI:10.3390/sym16070855
Ruze Yin; Jiaxin Mu; Takao Komatsu The p-Frobenius Number for the Triple of the Generalized Star Numbers, Symmetry, Volume 16 (2024) no. 8, p. 1090 | DOI:10.3390/sym16081090
Takao Komatsu; Claudio Pita-Ruiz The Frobenius Number for Jacobsthal Triples Associated with Number of Solutions, Axioms, Volume 12 (2023) no. 2, p. 98 | DOI:10.3390/axioms12020098
Takao Komatsu; Shanta Laishram; Pooja Punyani p-Numerical Semigroups of Generalized Fibonacci Triples, Symmetry, Volume 15 (2023) no. 4, p. 852 | DOI:10.3390/sym15040852
Takao Komatsu; Haotian Ying The p-Numerical Semigroup of the Triple of Arithmetic Progressions, Symmetry, Volume 15 (2023) no. 7, p. 1328 | DOI:10.3390/sym15071328

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