Find all odd numbers $n$ such that $q= \frac{\ln(3n+1)}{6\ln(2)}$ is also an odd number.

Question

Find all odd numbers $n$ such that

$$ q= \dfrac{\ln(3n+1)}{6\ln(2)}$$ is also an odd number.

Answer 1

The equation is equivalent to: $$q=\log_{64}(3n+1)$$ or $$n=\frac{64^q-1}3$$

Since $64\equiv 1\pmod 3$, $n$ is an integer. And since $64^q-1$ is odd, $n$ is odd. So you only have to substitute $q$ for odd numbers, and you get the possible values of $n$.