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Combinatoire, Théorie des nombres
The Frobenius number associated with the number of representations for sequences of repunits
Takao Komatsu1 
1 Department of Mathematical Sciences, School of Science, Zhejiang Sci-Tech University, Hangzhou 310018 China
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 73-89.

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 ,⋯,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≥2) as a generalization of the result of p=0 in [16].

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DOI : 10.5802/crmath.394
Classification : 11D07, 05A15, 05A17, 05A19, 11B68, 11D04, 11P81

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},
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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/

[1] Roger Apéry Sur les branches superlinéaires des courbes algébriques, C. R. Acad. Sci. Paris, Volume 222 (1946), pp. 1198-1200 | Zbl

[2] Fabián Arias; Jerson Borja The Frobenius problem for numerical semigroup generated by sequences that satisfy a linear recurrence relation (2021) (https://arxiv.org/abs/2111.04899)

[3] Matthias Beck; Ira M. Gessel; Takao Komatsu The polynomial part of a restricted partition function related to the Frobenius problem, Electron. J. Comb., Volume 8 (2001) no. 1, 7, 5 pages | MR | Zbl

[4] Matthias Beck; Curtis Kifer An extreme family of generalized Frobenius numbers, Integers, Volume 11 (2011) no. 5, A24, pp. 639-645 | MR | Zbl

[5] Albert H. Beiler Recreations in the theory of numbers – the queen of mathematics entertains, Dover Publications, 1966 | Zbl

[6] Damanvir S. Binner The number of solutions to ax+by+cz=n and its relation to quadratic residues, J. Integer Seq., Volume 23 (2020) no. 6, 20.6.5, 19 pages | MR | Zbl

[7] Alfred Brauer; James E. Shockley On a problem of Frobenius, J. Reine Angew. Math., Volume 211 (1962), pp. 215-220 | Zbl

[8] Arthur Cayley On a problem of double partitions, Philos. Mag., Volume XX (1860), pp. 337-341 | DOI

[9] Takao Komatsu On the number of solutions of the Diophantine equation of Frobenius-General case, Math. Commun., Volume 8 (2003) no. 2, pp. 195-206 | MR | Zbl

[10] Takao Komatsu The Frobenius number for sequences of triangular numbers associated with number of solutions, Ann. Comb., Volume 26 (2022) no. 3, pp. 757-779 | DOI | MR | Zbl

[11] Takao Komatsu Sylvester power and weighted sums on the Frobenius set in arithmetic progression, Discrete Appl. Math., Volume 315 (2022), pp. 110-126 | DOI | MR | Zbl

[12] Takao Komatsu; Yuan Zhang Weighted Sylvester sums on the Frobenius set, Ir. Math. Soc. Bull., Volume 87 (2021), pp. 21-29 | MR | Zbl

[13] Takao Komatsu; Yuan Zhang Weighted Sylvester sums on the Frobenius set in more variables, Kyushu J. Math., Volume 76 (2022) no. 1, pp. 163-175 | DOI | MR | Zbl

[14] Aureliano M. Robles-Pérez; José C. Rosales The Frobenius number for sequences of triangular and tetrahedral numbers, J. Number Theory, Volume 186 (2018), pp. 473-492 | DOI | MR | Zbl

[15] José C. Rosales; Manuel B. Branco; Denise Torrão The Frobenius problem for Thabit numerical semigroups, J. Number Theory, Volume 155 (2015), pp. 85-99 | DOI | MR | Zbl

[16] José C. Rosales; Manuel B. Branco; Denise Torrão The Frobenius problem for repunit numerical semigroups, Ramanujan J., Volume 40 (2016) no. 2, pp. 323-334 | DOI | MR | Zbl

[17] José C. Rosales; Manuel B. Branco; Denise Torrão The Frobenius problem for Mersenne numerical semigroups, Math. Z., Volume 286 (2017) no. 1-2, pp. 741-749 | DOI | MR | Zbl

[18] Ernst S. Selmer On the linear diophantine problem of Frobenius, J. Reine Angew. Math., Volume 293/294 (1977), pp. 1-17 | MR | Zbl

[19] James J. Sylvester On the partition of numbers, Quart. J., Volume 1 (1857), pp. 141-152

[20] Amitabha Tripathi The number of solutions to ax+by=n, Fibonacci Q., Volume 38 (2000) no. 4, pp. 290-293 | MR | Zbl

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