Question 41
Define a sequence $\left\langle a_{n}\right\rangle_{n=1}^{\infty}$ as follows:
$$
a_{n}= \begin{cases}0, & \text{ if the number of positive divisors of } n \text{ is odd } \\ 1, & \text{ if the number of positive divisors of } n \text{ is even }\end{cases}
$$
(The positive divisors of $n$ include 1 as well as $n$.) Let $x=0 . a_{1} a_{2} a_{3} \ldots$ be the real number whose decimal expansion contains $a_{n}$ in the $n$-th place, $n \geq 1$. Determine, with proof, whether $x$ is rational or irrational.
$$
a_{n}= \begin{cases}0, & \text{ if the number of positive divisors of } n \text{ is odd } \\ 1, & \text{ if the number of positive divisors of } n \text{ is even }\end{cases}
$$
(The positive divisors of $n$ include 1 as well as $n$.) Let $x=0 . a_{1} a_{2} a_{3} \ldots$ be the real number whose decimal expansion contains $a_{n}$ in the $n$-th place, $n \geq 1$. Determine, with proof, whether $x$ is rational or irrational.