On searching a table consistent with division poset

Suppose Pn = {1, 2, . . . , n} is a partially ordered set with the partial order defined by divisibility, that is, for any two distinct elements i, j ∈ Pn satisfying i divides j, i <Pn j. A table An = {ai|i = 1, 2, . . . , n} of distinct real numbers is said to be consistent with Pn, provided for any two distinct elements i, j ∈ {1, 2, . . . , n} satisfying i divides j, ai < aj . Given an real number x, we want to determine whether x ∈ An, by comparing x with as few entries of An as possible. In this paper we investigate the complexity τ(n), measured in the number of comparisons, of the above search problem. We present a 55n 72 + O(ln n) search algorithm for An and prove a lower bound ( 4 + 17 2160 )n+O(1) on τ(n) by using an adversary argument.