The Frobenius number for sequences of arithmetic progressions associated with number of solutions

Abstract

For given positive integers a 1 , a 2 , . . . , a k with gcd(a 1 , a 2 , . . . , a k ) = 1, consider the number of nonnegative solutions (x 1 , x 2 , . . . , x k ) of the linear equation a 1 x 1 + a 2 x 2 + • • • + a k x k = n for a positive integer n. For a given nonnegative integer p, there is a maximum n such that the number of nonnegative integer solutions is at most p, and it is very attractive to find the explicit formula of the maximum n. In fact, when p = 0, such a problem of finding the maximum integer n is called the linear Diophantine problem of Frobenius, and this maximum n is called the Frobenius number. The explicit formula for two variables is known not only for p = 0 but also for p > 0, but when there are three or more variables, it is difficult even in the special case of p = 0. For p > 0, it is not only more difficult, but no explicit formula has been found. In this paper, an explicit formula of such a generalized Frobenius number is given for the sequence of arithmetic progressions, in particular, for three variables. Further, explicit formulas are given not only for the maximum value of n such that the number of solutions is at most p, but also for the total number, the sum, and the weighted sum of such solutions.