A number theory problem by Yash Singhal

Trevor creates an infinite sequence \(S\) of integers by repeatedly adding the digits \(2,0,1\) and \(3\) to the previous numbers in the sequence. For example, the first few numbers that he forms are: \(2,20,201,2013,20132,201320,2013201,20132013,201320132\) and so on. Find the number of primes in \(S\).

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