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if a_n <a_m then a_{kn} <a _{km}, consistent sequences

Source: Ukraine TST 2010 p8

May 5, 2020
algebraSequenceinequalities

Problem Statement

Consider an infinite sequence of positive integers in which each positive integer occurs exactly once. Let {an},n≥1\{a_n\}, n\ge 1 be such a sequence. We call it consistent if, for an arbitrary natural kk and every natural n,mn ,m such that an<ama_n <a_m, the inequality akn<akma_{kn} <a _{km} also holds. For example, the sequence an=na_n = n is consistent . a) Prove that there are consistent sequences other than an=na_n = n. b) Are there consistent sequences for which an≠n,n≥2a_n \ne n, n\ge 2 ? c) Are there consistent sequences for which an≠n,n≥1a n \ne n, n\ge 1 ?
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