Phi-Pi-Po-Pum

For any positive integer n,n, the Euler's totient function ϕ(n)\phi(n) is defined as the number of integers from 11 to nn that are coprime to n.n. We will call a positive integer mm infinitely Euler, if there exists an infinite sequence of positive integers m1,m2,m3,...m_1,m_2,m_3,..., such that m1=mm_1=m and mi=ϕ(mi+1)m_i=\phi(m_{i+1}) for all i1.i\geq 1. How many positive integers less than 10001000 are infinitely Euler?

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