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, Sierpiński 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.

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