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2019 CNMO P8

Source: 2019 China North MO

February 22, 2020
number theory

Problem Statement

For positive intenger nn, define f(n)f(n): the smallest positive intenger that does not divide nn. Consider sequence (an):a1=a2=1,an=af(n)+1(n≥3)(a_n): a_1=a_2=1, a_n=a_{f(n)}+1(n\geq3). For example, a3=a2+1=2,a4=a3+1=3a_3=a_2+1=2,a_4=a_3+1=3. (a) Prove that there exists a positive intenger CC, for any positive intenger nn, an≤Ca_n\leq C. (b) Are there positive intengers MM and TT, satisfying that for any positive intenger n≥Mn\geq M, an=an+Ta_n=a_{n+T}.
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