Prime number of divisors, where does one use this? I recently asked this question Total number of divisors is a prime.
How useful is this property? What are its applications?
 A: The property is as useful as to say that a given number $n$ is of the form $p^{q-1}$ for primes $p$ and $q$. Indeed, $\tau(n)$ is prime if and only if $n=p^{q-1}$. Of course, the arithmetic function $\tau (n)$ is important in number theory. Even more so the function $\sigma(n)$, the sum of all divisors. The Riemann hypothesis is equivalent to an elementary statement on $\sigma(n)$, see the paper of Lagarias (2002), "An elementary problem equivalent to the Riemann hypothesis".
A: I got this helpful answer at Ask Dr.Math so I'm posting it.
A number with just two divisors is prime, and primes are useful.  But is it useful to
characterize the numbers with exactly 17 divisors (for example)?  I don't know.
The concept that has applications is that of "multiplicative function".  In number theory, a
function f is called multiplicative if whenever m and n are relatively prime, f (mn)= f (m) f (n).
The function d (n) that counts the number of divisors of n is one example.
If f is multiplicative, f is completely determined once you know its values on prime powers.
That's the key thing we need to know to characterize the numbers n for which d (n) is
prime.  So in my opinion, the importance of this problem is that it helps get you more
familiar with the idea of a multiplicative function.  It's this concept that has the
applications. 
