MathDB

Let

(a_i)_{i\in \mathbb{N}}

and

(p_i)_{i\in \mathbb{N}}

be two sequences of positive integers such that the following conditions hold:

\bullet ~~a_1\ge 2

\bullet~~ p_n

is the smallest prime divisor of

a_n

for every integer

n\ge 1

\bullet~~ a_{n+1}=a_n+\frac{a_n}{p_n}

for every integer

n\ge 1

Prove that there is a positive integer

N

such that

a_{n+3}=3a_n

for every integer

n>N

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