Which numbers have prime number of divisors?
For example, $16$ has $1$, $2$, $4$, $8$, $16$, a total of $5$ divisors, $5$ being a prime.
I found that primes and the power of primes such that $p^{q-1}$, where $p$ and $q$ are prime numbers, all have prime number of divisors. Is this property
limited to only these numbers?
Best Answer
If $n=p_1^{k_1}p_2^{k_2}\cdots p_h^{k_h}$ for ($p_i$ prime), then the number of divisors will be $(k_1+1)(k_2+1)\cdots(k_h+1)$.
So you are almost right. You need only one prime $p_1$ in the factorization and its exponent $k_1$ must be a prime minus one.