On a problem of Sierpinski

Abstract

Let $s\ge 2$ be an integer. Denote by $\mu_s$ the least integer so that every integer $\ell >\mu_s$ is the sum of exactly $s$ integers $>1 $ which are pairwise relatively prime. In 1964, Sierpi\'nski asked a determination of $\mu_s$. Let $p_1=2$, $p_2=3, ...$ be the sequence of consecutive primes and let $\mu_s = p_2+p_3+...+p_{s+1}+c_s$. P. Erd\H os proved that there exists an absolute constant $C$ with $-2\le c_s\le C$. In this paper, we determine $\mu_s$ for all $s\ge 2$. As a corollary, we show that $-2\le c_s\le 1100$ and the set of integers $s$ with $\mu_s= p_2+p_3+... +p_{s+1}+1100$ has the asymptotic density 1.