Q3/011) Rational or Irrational

Let a_n be defined as follows for all natural numbers n:
a_n = 0 if the number of divisors of n (including 1 and n) is odd
a_n = 1 otherwise.
Now consider the fraction 0.a₁a₂a₃....
Is this fraction rational or irrational? Explain.

Solution

The number of factors is odd for square number and it is even for non square numbers.
so a_n = 0 for no square number and = 1 for no square number
so in the decimal at each square place it is 1 and the gaps keeps on increasing .
so the digits do not recur and hence it is irrational