# Let $d(n)$ be the number of positive divisors of $n$. Find all $n$ such that $\frac{n}{d(n)}$ is prime.

Let $$d(n)$$ be the number of positive divisors of $$n$$. Find all $$n$$ such that $$\frac{n}{d(n)}=p$$, a prime.

If $$n=\prod_{1\leq i \leq k} p_i^{r_i}$$, then $$\begin{eqnarray*} n&=&p_1\cdot d(n)=p_1(r_1+1)(r_2+1)...(r_k+1)\\ &\ & \implies p_1^{r_1-1}p_2^{r_2}...p_k^{r_k}=(r_1+1)(r_2+1)...(r_k+1). \end{eqnarray*}$$ But I do not get any clue here. (I previously posed a question which asked only for primes but that got only one answer, which was just an observation. But now i have deleted that account of mine and have no means for it to gain attention). Can you please help me? Thanks!

• Good start! Can you show from your identity (or otherwise) that $p_1^{r_1-1} \le r_1+1$? Can you find all solutions to that inequality? Commented Apr 25, 2019 at 16:55
• Only possibility i guess occurs in $r_1=2, p_1=3$, et cetera
– user665856
Commented Apr 25, 2019 at 16:56

Let $$n=\prod_{i=1}^kp_i^{r_i}$$ be given, where the $$p_i$$ are distinct primes and the $$r_i$$ are positive integers. You're off to a good start with the identity $$d(n)=\prod_{i=1}^k(r_i+1).$$ From this it follows that $$\frac{n}{d(n)}=\frac{\prod_{i=1}^kp_i^{r_i}}{\prod_{i=1}^k(r_i+1)}=\prod_{i=1}^k\frac{p_i^{r_i}}{r_i+1}.$$ It is clear that if $$\tfrac{n}{d(n)}=p$$ is prime then $$p\mid n$$, so without loss of generality $$p=p_1$$. Then $$1=\frac{n}{p_1d(n)}=\frac{p_1^{r_1-1}}{r_1+1}\prod_{i=2}^k\frac{p_i^{r_i}}{r_i+1}.$$ Now prove by induction that
For all primes $$p$$ and all positive integers $$r$$ you have $$\tfrac{p^r}{r+1}\geq 1$$, with equality iff $$(p,r)=(2,1)$$.