Is there a positive integer $n \ge 2$ for which $\frac{k}{\pi(k)} = n$ has no solution?

Question

For a given positive integer $n \ge 2$ let $a_n$ be the number of integers $k$ such that $\dfrac{k}{\pi(k)} = n$ where $\pi(x)$ is the prime counting function. The first few values of $(n,a_n)$ are

$$(2, 4), (3, 3), (4, 3), (5, 6), (6, 7), (7, 6), (8, 6), (9, 3), (10, 9)(11, 1), (12, 18),$$ $$(13, 11),(14, 12),(15, 21),(16, 3),(17, 10), (18, 33), (19, 31), (20, 32), (21, 24)$$

In the example above we see that for $n \ge 2, a_n \ge 1$. Intuitively this is expected because by the prime number theorem, $\dfrac{k}{\pi(k)} \sim \log k$. Hence as $k$ increases, the integer part of $\log k$ is expected to run through all positive integers after a certain point. However $a_k$ we notice that is not strictly increasing and $a_{11} = 1$. This brings the question:

Question: Is there a positive integer $n \ge 2$ for which $\dfrac{k}{\pi(k)} = n$ has no solution?

Answer 1

The answer is no, there is no such $n$. Observe that the fraction increases if and only if $\pi (k)$ stagnates. If $\frac{k}{\pi (k)} \leq n < \frac{k+1}{\pi (k+1)}$, then $\pi (k+1)=\pi (k)$ and thus $n\pi (k) -1<k\leq n\pi (k)$ with $k\in \mathbb{N}$, i.e. $k=n\pi (k)$.

Answer 2

Not an answer but too long for comment: here is a Mathematica script to look for solutions systematically:

solve[] := Module[
   {i, n},
   i = 2;
   n = 2;
   While[True,
    While[i/PrimePi[i] != n,
     i++
     ];
    Print["solve(", n, ")=", i];
    n++;
    ];
   ];

The first few solutions for $n=2,3,4,...$

solve(2)=2
solve(3)=27
solve(4)=96
solve(5)=330
solve(6)=1008
solve(7)=3059
solve(8)=8408
solve(9)=23526
solve(10)=64540
solve(11)=175197
solve(12)=480852
solve(13)=1304498
solve(14)=3523884
solve(15)=9557955
solve(16)=25874752
solve(17)=70115412

Answer 3

Solving in pari/gp:

? #
   timer = 1 (on)
?
? k=2;for(n=2,20,while(k/primepi(k)!=n,until(z=k;k=n*primepi(k),z!=k));print1(k", "))
2, 27, 96, 330, 1008, 3059, 8408, 23526, 64540, 175197, 480852, 1304498, 3523884, 9557955, 25874752, 70115412, 189961182, 514272411, 1394193580,
time = 9min, 13,087 ms.