We denote the prime-counting function with $\pi(x)$ and we consider integer solutions $n\geq 1$ of the congruence $$2^{\pi(n)}\equiv 1\text{ mod }n.\tag{1}$$ Then the sequence of solutions starts as $$1,3,39,43,63,91\ldots$$
Question. I would like to know if it is possible to get an approximation for the number of solutions of $(1)$ when $n$ runs over the segment of integers $1\leq n\leq N$ ( we assume that for some large integer $N$). What work can be done? Many thanks.
You can compute more solutions using this code
for (n = 1, 10000,if (Mod(2^(primepi(n)),n)==1,print(n)))
from this Sage Cell Server (choose GP as language and press Evaluate).
The $\color\red {473}$ solutions upto $n=10^7$ are :
The solutions upto $10^9$ that are prime are :