How many solutions do exist for $\varphi(n)=\pi(n), n\in\mathbb{N}$?

Question

How many solutions do for the following equation exists?

\begin{align} \varphi(n)=\pi(n), n\in\mathbb{N}, \end{align}

where $\varphi(n)$ is Euler's totient function and $\pi(n)$ is the prime-counting function.

Thank you very much

Answer 1

Surprisingly, the list of solutions is finite: 2, 3, 4, 8, 10, 14, 20, 90. Moreover $\varphi(n)>\pi(n)$ for $n>90$.

A proof is given at page 179 in the following paper by Leo Moser: "On the equation $\varphi(n)=\pi(n)$", Pi Mu Epsilon J. 1951, 177–180.

Sketch of Moser's proof. We have that $\varphi(n)-\pi(n)=B(n) - A(n)$ where $A(n)$ is the number of prime divisors of $n$ and $B(n)$ is the number of non-primes, which do not exceed $n$ and are relatively prime to $n$.

Now i) $\pi(\sqrt{n})\geq 2A(n)$ for $n>360$ (lemma 3 where Bertrand's postulate is used) and ii) $B(n)>\pi(\sqrt{n})-A(n)$ (lemma 4).

Hence by for $n>360$, $$\varphi(n)-\pi(n)=B(n) - A(n)>\pi(\sqrt{n})-A(n)-A(n)\geq 0.$$

P.S. According http://oeis.org/A037171, the result has been proved by David W. Wilson and Jeffrey Shallit, but unfortunately no reference is provided.