On a problem of Sierpinski

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

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