$p$-numerical semigroup of the sequence of consecutive integers
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For a set of positive integers $A=\{a_1,a_2,\dots,a_k\}$ with \linebreak $\gcd(a_1,a_2,\dots,a_k)=1$, let $S_p$ be the set of non-negative integers that can be represented by the linear combinations of non-negative integral coefficients of $a_1,a_2,\dots,a_k$ in more than $p$ ways. The main interests are on explicit closed formulas for the largest element $g_p(A)$, the cardinality $n_p(A)$ and the sum of the elements $s_p(A)$ of $\mathbb N_0\backslash S_p$. When $p=0$, the problem to study these values is the well-known and famous linear Diophantine problem of Frobenius. The general formulas are given when $k=2$, and some formulas are given for only special cases when $k=3$. But when $p>0$ and $k\ge 4$, no concrete explicit formula had been found. In this paper, we are successful in showing explicit formulas when the elements in $A$ consist of the sequence of consecutive integers $a,a+1,\dots, 2 a-1$.
Abstract
For a set of positive integers $A=\{a_1,a_2,\dots,a_k\}$ with \linebreak $\gcd(a_1,a_2,\dots,a_k)=1$, let $S_p$ be the set of non-negative integers that can be represented by the linear combinations of non-negative integral coefficients of $a_1,a_2,\dots,a_k$ in more than $p$ ways. The main interests are on explicit closed formulas for the largest element $g_p(A)$, the cardinality $n_p(A)$ and the sum of the elements $s_p(A)$ of $\mathbb N_0\backslash S_p$. When $p=0$, the problem to study these values is the well-known and famous linear Diophantine problem of Frobenius. The general formulas are given when $k=2$, and some formulas are given for only special cases when $k=3$. But when $p>0$ and $k\ge 4$, no concrete explicit formula had been found. In this paper, we are successful in showing explicit formulas when the elements in $A$ consist of the sequence of consecutive integers $a,a+1,\dots, 2 a-1$.