5

Part of 2012 Canadian Mathematical Olympiad Qualification Repechage

Problems(1)

Terms in a sequence divisible by 3^2011

Source: Canadian Repêchage 2012: Problem 5

Given a positive integer

n

d(n)

be the largest positive divisor of

n

n

d(8) = 4

d(13) = 1

. A sequence of positive integers

a_1, a_2,\dots

a_{i+1} = a_i +d(a_i),

for all positive integers

i

. Prove that regardless of the choice of

a_1

, there are inﬁnitely many terms in the sequence divisible by

3^{2011}

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