<s>[INST] Find at least $5$ integers $n$ such that $\varphi(n)=16$ 
Let $\varphi(n)$ denote Euler's totient function. Find all integers such that $\varphi(n)=16$. 

Answers given were $17,32,34,40,48.$ 
I am thinking a generalisation of this problem: is there a way to find all positive integers $n$ such that $\varphi(n)=k$ for a specific $k$? Is there a way to do it other than trial and error? (which is what I did BTW)
 [/INST] If $$n= p_1^{a_1}p_2^{a_2}...p_k^{a_k}$$  where $a_i\geq 1$ and $p_1<p_2<...<p_k$ then we have $$\phi(n) = p_1^{a_1-1}p_2^{a_2-1}...p_k^{a_k-1} (p_1-1)...(p_k-1)$$
$$16= p_1^{a_1-1}p_2^{a_2-1}...p_k^{a_k-1} (p_1-1)...(p_k-1)$$

If $p_1=2$ and $k>1$ $$2^4= 2^{a_1-1}p_2^{a_2-1}...p_k^{a_k-1} (p_2-1)...(p_k-1)$$
then $a_i=1$ for all $i>1$, so we have $$2^4= 2^{a_1-1} (p_2-1)...(p_k-1)$$
Now if $a_1 = 1$ then $k\leq 5$  so $$ 2^4= (p_2-1)...(p_k-1)$$
and thus $p_k\leq 17$. 
We see that $k=2$ and $p_2 = 17$ does works, so $\boxed{ n_1=34}$
and it is the only one if $a_1=1$
If $a_1 = 2$ then $k\leq 4$  so $$ 2^3= (p_2-1)...(p_k-1)$$
and thus $p_k\leq 7$. We see that $p_2 =3$ and $p_3=5$ works, so $\boxed{ n_2 = 60}$ 
If $a_1 = 3$ then $k\leq 3$  so $$ 2^2= (p_2-1)...(p_k-1)$$
and thus $p_k\leq 5$. We see that $p_2 =5$ works, so $\boxed{ n_3 = 40}$ 
If $a_1 = 4$ then $k=2$  so $$ 2= p_2-1$$
so $p_2 =3$ works, so $\boxed{n_4 = 48}$ 
If $a_1 =5$ then o $\boxed{n_5 = 32}$  and we have 5 numbers

If $p_1 >2$ then ...
</s>