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 $a_1,a_2,\ldots ,a_k$ 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.