Pi chains.

The prime counting function, denoted by \(\pi(x)\) counts the number of primes less or equal to \(x\).

Consider the process of continually finding the value of \(\pi(x)\) of \(x\) until it equals zero.

For example,for \(x=1000\) we have \[\pi(1000) = 168 \rightarrow 39 \rightarrow 12 \rightarrow 5 \rightarrow 3 \rightarrow 2 \rightarrow 1 \rightarrow 0\]

\[\pi(20) = 8 \rightarrow 4 \rightarrow 2 \rightarrow 1 \rightarrow 0 \]

From the above we can say that \(1000\) has a pi chain of length 8 and \(20\) has a pi chain of length 5.

How many integers have a pi chain of length 9?

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