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N consecutive terms-Iran 3rd round-Number Theory 2007

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July 28, 2010
number theory unsolvednumber theory

Problem Statement

For a positive integer nn, denote rad(n)rad(n) as product of prime divisors of nn. And also rad(1)=1rad(1)=1. Define the sequence {ai}i=1∞\{a_i\}_{i=1}^{\infty} in this way: a1∈Na_1 \in \mathbb N and for every n∈Nn \in \mathbb N, an+1=an+rad(an)a_{n+1}=a_n+rad(a_n). Prove that for every N∈NN \in \mathbb N, there exist NN consecutive terms of this sequence which are in an arithmetic progression.
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