On a problem of Sierpiński,Ⅱ

For any integer s ≥ 2, let μs be the least integer so that every integer l 〉 μs is the sum of exactly s integers which are pairwise relatively prime. In 1964, Sierpifiski asked for the determination of μs. Let Pi be the i-th prime and let μs = p2 ＋P3 ＋ … ＋ps＋1＋ cs. Recently, the authors solved this problem. In particular, we have （1） cs = -2 if and only if s = 2; （2） the set of integers s with cs= 1100 has asymptotic density one; （3） cs ∈ A for all s ≥ 3, where A is an explicit set with A [2, 1100] and ｜A｜ = 125. In this paper, we prove that, （1） for every a ∈ A, there exists an index s with cs = there are infinitely many s with es = a. We also point out can be applied to this problem. a; （2） under Dickson＇s conjecture, for every a∈ A, that recent progress on small gaps between primes

For any integer s≥ 2, let μsbe the least integer so that every integer l > μs is the sum of exactly s integers which are pairwise relatively prime. In 1964, Sierpi′nski asked for the determination of μs. Let pibe the i-th prime and let μs= p2 + p3 + + ps+1+ cs. Recently, the authors solved this problem. In particular,we have(1) cs=-2 if and only if s = 2;(2) the set of integers s with cs= 1100 has asymptotic density one;(3) cs∈ A for all s ≥ 3, where A is an explicit set with A ■[2, 1100] and |A| = 125. In this paper, we prove that,(1) for every a ∈ A, there exists an index s with cs= a;(2) under Dickson's conjecture, for every a ∈ A,there are infinitely many s with cs= a. We also point out that recent progress on small gaps between primes can be applied to this problem.