Factorial Divisibility (E)

For how many natural numbers \( n < 1000 \) does \( n \not | (n-1)! \)?

Details and Assumptions

  • \( n! \) is the factorial function where \( n! = n*(n-1)! \) and \( 0! = 1 \).

  • \( a \not | b \) means that \( a \) does not perfectly divide \(b \), i.e. \( a \) is not a divisor of \( b \).For example, \( 20 \not | 75 \).

  • You may use the List of Primes as a reference.

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