A Rocket of Primes

Number Theory Level pending

\[\large n \qquad\qquad \large n^2 + 4\qquad\qquad \large 2n^2 + 5n + 8 \\ \large 12n + 13\qquad\qquad \large 24n - 11\qquad\qquad \large n^2 + 398n +12\]

Let \(n\) be a positive integer such that the 6 numbers above are all prime numbers. Find the sum of all the possible values of \(n\).

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