I wanna know counter example of this infinite sequence statement

Question

if ${a_{kn}}$ coverges for all integer $2\leq k$, then sequence $a_{n}$ converges.($n=1,2,3...$)

This infinite sequence statement is maybe False. I'm trying to find counter example of this. Is there counter example of this statement?

(This is from Stewart's Calculus: Early Transcendentals)

Answer 1

One counterexample (due to Euclid...!) is $\displaystyle a_n = \begin{cases} 1, &\text{if $n$ is prime}, \\ 0, &\text{if $n$ is not prime}. \end{cases} $