Affiliation:
1. Department of Mathematical Sciences, School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China
Abstract
For given positive integers a1,a2,⋯,ak with gcd(a1,a2,⋯,ak)=1, the denumerant d(n)=d(n;a1,a2,⋯,ak) is the number of nonnegative solutions (x1,x2,⋯,xk) of the linear equation a1x1+a2x2+⋯+akxk=n for a positive integer n. For a given nonnegative integer p, let Sp=Sp(a1,a2,⋯,ak) be the set of all nonnegative integer n’s such that d(n)>p. In this paper, by introducing the p-numerical semigroup, where the set N0\Sp is finite, we give explicit formulas of the p-Frobenius number, which is the maximum of the set N0\Sp, and related values for the triple of arithmetic progressions. The main aim is to determine the elements of the p-Apéry set.
Subject
Physics and Astronomy (miscellaneous),General Mathematics,Chemistry (miscellaneous),Computer Science (miscellaneous)
Reference38 articles.
1. Komatsu, T., and Ying, H. (2024). p-numerical semigroups with p-symmetric properties. J. Algebra Appl., 23, (Online Ready).
2. Mathematical questions with their solutions;Sylvester;Educ. Times,1884
3. On subinvariants, i.e., semi-invariants to binary quantics of an unlimited order;Sylvester;Am. J. Math.,1882
4. A remark related to the Frobenius problem;Brown;Fibonacci Quart.,1993
5. A note on Brown and Shiue’s paper on a remark related to the Frobenius problem;Fibonacci Quart.,1994