Definition
A Fortunate number is the smallest integer m > 1 such that for a given positive integer n, primorial(n) + m is a prime number, where primorial(n) is the product of the first n prime numbers.

For example the first fortunate number is 3 because primorial(1) is 2 and 2 + 3 = 5 which is prime whereas 2 + 2 = 4 is composite.



Task
After sorting and removal of any duplicates, compute and show on this page the first 8 Fortunate numbers or, if your language supports big integers, the first 50 Fortunate numbers.



Related task
 Primorial numbers


See also
 oeis:A005235 Fortunate numbers 
 oeis:A046066 Fortunate numbers, sorted with duplicates removed


using Primes
 
primorials(N) = accumulate(*, primes(N), init = big"1")
 
primorial = primorials(800)
 
fortunate(n) = nextprime(primorial[n] + 2) - primorial[n]
 
println("After sorting, the first 50 distinct fortunate numbers are:")
foreach(p -> print(rpad(last(p), 5), first(p) % 10 == 0 ? "\n" : ""),
    (map(fortunate, 1:100) |> unique |> sort!)[begin:50] |> enumerate)
 