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a_n= a_{n-1} + a_{[n/3]}

Source: Vietnam TST 2001 for the 42th IMO, problem 1

June 26, 2005
calculusintegrationfloor functionalgebra unsolvedalgebra

Problem Statement

Let a sequence of integers {an}\{a_n\}, n∈Nn \in \mathbb{N} be given, defined by a0=1,an=an−1+a[n/3]a_0 = 1, a_n= a_{n-1} + a_{[n/3]} for all n∈N∗n \in \mathbb{N}^{*}. Show that for all primes p≤13p \leq 13, there are infinitely many integer numbers kk such that aka_k is divided by pp. (Here [x][x] denotes the integral part of real number xx).
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