Have I found all the numbers less than 50,000 with exactly 11 divisors?

The math problem I am trying to solve is to find all positive integers that meet these two conditions:

• have exactly 11 divisors
• are less than 50,000

My starting point is a number with exactly 11 divisors is of the form:

$c_1p_1 * c_2p_2 * c_3p_3 * ... * c_{11}p_{11}$

where: $c_i$ is an integer and $p_i$ is prime

and noting that 1 can be a divisor but can only be included once in the list of divisors.

Therefore the smallest such number is: $1*2^{10} = 1024$

The next such number would be: $2^{11} = 2048$

So far $c_i$ has been 1, but I will now start letting some of them be 2 until I reach the upper bound of 50,000. This gives me:

$2^{12}$, $2^{13}$, $2^{14}$ and $2^{15}$ (as $2^{16} > 50,000)$

I feel like I'm onto a useful pattern. So I will start using 3s as well (again until I reach the upper bound):

$3*2^{10}$, $3^2*2^9$, $3^3*2^8$, $3^4*2^7$, $3^5*2^6$, $3^6*2^5$, $3^7*2^4$

Now 4s: $4*2^{10}$, $4^2*2^9$, $4^3*2^8$, $4^4*2^7$

Then 5s: $5*2^{10}$, $5^2*2^9$, $5^3*2^8$

Then 6s: $6*2^{10}$, $6^2*2^9$

Then 7s: $7*2^{10}$, $7^2*2^9$

Then 8s: $8*2^{10}$, $8^2*2^9$

Then 9s: $9*2^{10}$, $9^2*2^9$

Then 10s: $10*2^{10}$

I've now reached the point where there's only one number in the sub-sequence. Therefore, I also have:

$11*2^{10}, 12*2^{10}, 13*2^{10}, ..., 48*2^{10}$

I now feel I've exhausted my algorithm. However I'm not sure if my answer is correct and complete. So, have I enumerated all such numbers? Or have I missed some or doubled-up?

If you're looking at a number with prime factorization $$ p_1^{a_1}p_2^{a_2}\cdots p_n^{a_n} $$ (note, there are no $c$s in a prime factorization), then it has exactly $(a_1+1)\cdot(a_2+1)\cdots(a_n+1)$ divisors.

Since $11$ is prime, the only way it can be written as such a product is just $(10+1)$. So you're looking for numbers of the form $p^{10}$ -- and only that form -- less than $50,000$.

If $n$ has a prime factorization of $p_1^ip_2^jp_3^k$, it has $(i+1)(j+1)(k+1)$ divisors. The numbers with exactly 11 divisors are of the form $p^{10}$ since 11 is prime.

Since $3^{10} > 50,000$, the only answer is $1024$.