An extremal problem in subsequence sum

摘要

Let N denote the set of all positive integers and let N0=N⋃{0}. For a strictly increasing sequence A of positive integers, let P(A) be the set of all integers which can be represented as the finite sum of distinct terms of A. Fix an integer b such that b∈{1,2,4,7,8} or b≥11. For every integer k≥1, define inductively ck(b) as the smallest positive integer r so that there exist two strictly increasing sequences A={ai}i=1∞ and B={bi}i=1∞ of positive integers such that (1) b1=c1(b)=b,b2=c2(b)=3b+5; (2) bi=ci(b) for all 3≤i≤k−1 and bk=r; (3) P(A)=N0∖{bi:i∈N} and ai≤∑j