Prime factors?

All integers can be written uniquely as the product of their prime factors:

$n=A^a \times B^b \times C^c \times ...$

Where $A,B,C,...$ are prime numbers, and $a,b,c...\geq 1$ .

This number, $n$ , has $(a+1)(b+1)(c+1)...$ factors in total.

As $a,b,c...\geq 1$ , $(a+1),(b+1),(c+1)...\geq 2$ .

We can see from this that if a number has more than one prime factor it must have a composite number of factors, as its total number of factors will be the product of two or more numbers.

We are therefore looking for numbers with only one prime factor, i.e. $n=p^y$ , where $p$ is a prime, and $y \geq 2$ , as the question excludes prime numbers.

These numbers, $n=p^y$ , have $y+1$ factors.

We are therefore looking for primes raised to powers which are one less than a prime number, eg. $2,4,6,10...$ . The result of this must also be less than $1000$ :

From this table, we can see that our answer is $\boxed{16}$ non-prime numbers below $1000$ with a prime number of factors.