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a_{n+1} = a_n + \delta_(a_n) where \delta_(a) =second largest divisor of a

Source: Switzerland - 2022 Swiss Final Round p5

November 17, 2022

number theoryrecurrence relationdivisor

Problem Statement

For an integer

a \ge 2

, denote by

\delta_(a)

the second largest divisor of

a

. Let

(a_n)_{n\ge 1}

be a sequence of integers such that

a_1 \ge 2

and

a_{n+1} = a_n + \delta_(a_n)

for all

n \ge 1

. Prove that there exists a positive integer

k

such that

a_k

is divisible by

3^{2022}