Let N denote the set of all positive integers and let N-0 = N boolean OR{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 is an element of {1, 2, 4, 7, 8} or b >= 11. For every integer k >= 1, define inductively c(k)((b)) as the smallest positive integer r so that there exist two strictly increasing sequences A = {a(i)}(i=1)(infinity) and B = {b(i)}(i=1)(infinity) 1 of positive integers such that (1) b1 = c(1)((b)), b(2) = c(2)((b))= 3b + 5; (2) bi = c(i)((b)) for all 3 <= i <= k - 1 and b(k) = r; (3) P(A) = N-0 \ {b(i) : i is an element of N} and a(i) <= Sigma(j<i) a(j) + 1 for all a(i) >= b + 1. Furthermore, let C(b) := {C-(b) : {c(k)((b))k is an element of N} and A((b)) := : A is a strictly increasing sequence of positive integers such that P(A) = N-0 \ C-(b). For any positive real number x, we let A(x) denote the number of terms no more than x in the sequence A and define L-x((b)) := min(A is an element of A(b)) A(x). In this paper, we determine the exact value of L-x((b)) for any positive real number x, and give a characterization on the strictly increasing sequence A of positive integers such that P(A) = N-0 \C-(b) and A(x) = L-x((P)) for any positive real number x >= c(1)((b)) -1. (C) 2022 Elsevier Inc. All rights reserved.
Subsequence sumBurr's problemComplementExtremal problemINVERSE PROBLEM
NSTL
Elsevier
