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Vadym Koval is back. Though he never left

Source: Ukrainian Mathematical Olympiad 2021. Day 2, Problem 9.8

December 21, 2023

number theory

Problem Statement

For any positive integer

a > 1

, we define the sequence

(a_n)

as follows:

a_{n+1} = a_n + d(a_n) - 1

n \in \mathbb{N}

a_1 = a

, where

d(b)

denotes the smallest prime divisor of

b

. Prove that for any positive integer

k

, the sequence

d(a_n)

for

n \geq k

is not increasing, i.e. the condition

d(a_{n+1}) > d(a_n)

is not true for at least one

n \geq k

.
Proposed by Vadym Koval