On a problem of Sierpinski

TL;DR

This paper determines the exact values of the minimal integer micable to represent all integers greater than it as sums of pairwise coprime integers, extending Sierpi
4dnski's 1964 problem and analyzing the asymptotic density of certain cases.

Contribution

It explicitly calculates micable for all s  2, resolving a longstanding problem posed by Sierpi
4dnski and refining bounds on the associated constants.

Findings

micable values are explicitly determined for all s  2.

The bounds on c_s are tightened to -2  c_s  1100.

The set of s with micable = sum of specific primes plus 1100 has density 1.

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.