Phi-Pi-Po-Pum

Number Theory Level 5

For any positive integer $n,$ the Euler's totient function $\phi(n)$ is defined as the number of integers from $1$ to $n$ that are coprime to $n.$ We will call a positive integer $m$ infinitely Euler, if there exists an infinite sequence of positive integers $m_1,m_2,m_3,...$ , such that $m_1=m$ and $m_i=\phi(m_{i+1})$ for all $i\geq 1.$ How many positive integers less than $1000$ are infinitely Euler?