We study some arithmetic properties of the Ramanujan function $\tau(n)$, such as the largest prime divisor $P(\tau(n))$ and the number of distinct prime divisors $\omega(\tau(n))$ of $\tau(n)$ for various sequences of $n$. In particular, we show that \hbox{ $P(\tau(n)) \geq (\log n)^{ 33/31 + o(1)}$} for infinitely many $n$, and \begin{ equation*} P(\tau(p)\tau(p^2)\tau(p^3)) > (1+o(1))\frac{ \log\log p\log\log\log p} { \log\log\log\log p} \end{ equation*} for every prime $p$ with \hbox{ $\tau(p)\neq 0$}.
Categories: math.NT
