On a problem of Sierpiński, II

Abstract

For any integer s ⩾ 2, let µ s be the least integer so that every integer l > µ s is the sum of exactly s integers which are pairwise relatively prime. In 1964, Sierpinski asked for the determination of µ s . Let p i be the i-th prime and let µ s = p 2 + p 3 + ⋯ + p s+1 + c s . Recently, the authors solved this problem. In particular, we have (1) c s = −2 if and only if s = 2; (2) the set of integers s with c s = 1100 has asymptotic density one; (3) c s ∈ 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 c s = a; (2) under Dickson’s conjecture, for every a ∈ A, there are infinitely many s with c s = a. We also point out that recent progress on small gaps between primes can be applied to this problem.