On a problem of J. H. Fang and Z. K. Fang

Let A be a sequence of positive integers and P(A) be the set of all integers which are the finite sum of distinct terms of A . Let B = { b 1 < b 2 < ⋯ } be a sequence of positive integers with b 1 ∈ { 4 , 7 , 8 } ∪ [ 11 , + ∞ ) . By observing the results of Chen and Fang [2], Fang and Fang [4] found that 3 b 1 + 5 is the critical value for b 2 such that there exists a sequence A of positive integers for which P ( A ) = N \ B . For b 2 = 3 b 1 + 5 , they considered the inverse problem and determined the critical value for b 3 . Also, for b 2 ≥ 3 b 1 + 5 , they asked for the critical value for b 3 . In this paper, we solve this problem.