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Increasing sequence, Decreasing Euler's totient function

Source: Iran 3rd round 2012-Final exam-P3

September 24, 2012
functionceiling functionlogarithmsinductionnumber theory proposednumber theory

Problem Statement

Prove that for each n∈Nn \in \mathbb N there exist natural numbers a1<a2<...<ana_1<a_2<...<a_n such that ϕ(a1)>ϕ(a2)>...>ϕ(an)\phi(a_1)>\phi(a_2)>...>\phi(a_n).
Proposed by Amirhossein Gorzi
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