Anjans A.

$a_{1}, a_{2}, a_{3}, \ldots$ - an increasing sequence of natural numbers. It is known that $a_{a_{k}}=3 k$ for any $k$.

Find
a) $a_{100}$

b) $a_{1983}$.
a) Immediately, we notice that the sequence $a_{k}$ is strictly increasing. Indeed, the assumption $a_{k}=a_{k+1}=n$ immediately leads to a contradiction: $a_{n}=3 k=3(k+1)$.

Moreover, $a_{1}>1$ (otherwise, $a_{a_{1}}=a_{1}=1 \neq 3$). From this, it follows that $a_{k}>k$ for all $k$. On the other hand, $a_{1}<a_{a_{1}}=3$. Therefore, $a_{1}=2$,

$a_{2}=3, a_{3}=6, a_{6}=9, a_{9}=18, a_{18}=27, a_{27}=54, a_{54}=81, a_{81}=162, a_{162}=243$. Since $162-81=243-162$, it follows that $a_{k}=81+k$ for all $k$ from 81 to 162. In particular, $a_{100}=181$.

b) $a_{162}=243, a_{243}=486, a_{486}=729$. Since $729-486=486-243$, it follows that $a_{k}=243+k$ for all $k$ from 243 to 486. In particular, $a_{418}=661$, and thus $a_{661}=1254, a_{1254}=1983, a_{1983}=3762$.

## Answer

a) 181;
b) 3762.