Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 11H / c

Let $P=p_1{p}_2\cdots p_{k+1}$ be the product of the first $k+1$ primes, and take $n=P^a$ for positive integers $a$. The prime factorization givesTheir ratio isThus every sufficiently large $a$ gives $\tau (P^a)\ge (\log P^a{)}^k$, producing infinitely many such $n$.