On Frobenius problem with restrictions on common divisors of coefficients

Abstract

Let $m,s,t$ are positive integers with $t\leq s-2$ and $a_1,a_2,\ldots,a_s$ are positive integers such that $(a_1,a_2,\ldots,a_{s-1})=1$. In the paper we prove that every sufficiently large positive integer can be written in the form $a_1\mu_1+a_2\mu_2+\ldots+a_s\mu_m$, where positive integers $\mu_1,\mu_2,\ldots,\mu_s$ have no common divisor being $m$-th power of a positive integer greater than $1$ but each $t$ of the values of $\mu_1,\mu_2,\ldots,\mu_n$ 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 $\mu_1,\mu_2,\ldots,\mu_n$ with no common divisor being $m$-th power of a positive integer greater than $1$ but each $s-1$ of the values of $\mu_1,\mu_2,\ldots,\mu_s$ have a common divisor being $m$-th power of a positive integer greater than $1$.