On Frobenius problem with restrictions on common divisors of coefficients

Abstract

Let $m,s,t$ are positive integers with $t\le s-2$ and $a_1,a_2,\dots ,a_s$ are positive integers such that $(a_1,a_2,\dots ,a_{s-1})=1$. In the paper we prove that every sufficiently large positive integer can be written in the form $a_1{\mu }_1+a_2{\mu }_2+\dots +a_s{\mu }_m$, where positive integers ${\mu }_1,{\mu }_2,\dots ,{\mu }_s$ have no common divisor being $m$-th power of a positive integer greater than $1$ but each $t$ of the values of ${\mu }_1,{\mu }_2,\dots ,{\mu }_n$ have a common divisor being $m$-th power of a positive integer greater than $1$. Moreover, we show that every sufficiently large positive integer can be written as a sum of positive integers ${\mu }_1,{\mu }_2,\dots ,{\mu }_n$ with no common divisor being $m$-th power of a positive integer greater than $1$ but each $s-1$ of the values of ${\mu }_1,{\mu }_2,\dots ,{\mu }_s$ have a common divisor being $m$-th power of a positive integer greater than $1$.

14 pages