Phi-Pi-Po-Pum

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?

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