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Set of divisors of a recurrent sequence

Source: 4th German TST 2005, problem 3

June 3, 2005

inductionnumber theory proposednumber theory

Problem Statement

Let

b

and

c

be any two positive integers. Define an integer sequence

a_n

, for

n\geq 1

, by

a_1=1

a_2=1

a_3=b

and

a_{n+3}=ba_{n+2}a_{n+1}+ca_n

. Find all positive integers

r

for which there exists a positive integer

n

such that the number

a_n

is divisible by

r