I am interested in the function $f(n)$ defined on the positive integers where $f(n)$ is the smallest integer for which the number of divisors is the $n$-th prime. Is there a simple asymptotic form for this function?
The first few values of this function are $2, 4, 16, 64, 1024, 4096$.
If the number of divisors of some number n is a prime number, n must be a prime power. Therefore , $2^{p-1}$ is the least number having p divisors for any prime p.