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%I #35 Sep 08 2022 08:45:57
%S 0,2,3,5,6,18,44,113,2615,16914
%N Numbers k such that k! + 2*k + 1 is prime.
%C Some of the larger entries may only correspond to probable primes.
%e 6! + 2*6 + 1 = 733 is prime, so 6 is in the sequence.
%p select(p -> isprime(factorial(p) + 2*p + 1), [$0 .. 1000]) # _Reza K Ghazi_, Jul 11 2021
%t Select[Range[0, 1000], PrimeQ[#! + 2*# + 1] &] (* _Reza K Ghazi_, Jul 11 2021 *)
%o (Python)
%o factor=1;
%o for i in range(2,10000):
%o factor=factor*i
%o if is_probable_prime(factor+2*i+1): #A Miller-Rabin primality test, with k=10
%o print i
%o (Magma) [ n: n in [0..1000] | IsPrime(Factorial(n)+2*n+1) ]; // _Klaus Brockhaus_, Jul 02 2011
%o (PARI) for(n=0,10^5,N=n!+2*n+1;if(ispseudoprime(N),print1(n,", ")));
%o /* use isprime() for stricter checking */ /* _Joerg Arndt_, Jul 03 2011 */
%o (SageMath) [p for p in range(1000) if is_prime(factorial(p)+2*p+1)] # _Reza K Ghazi_, Jul 11 2021
%Y Cf. A073308 (k!+k+1 is prime).
%K hard,nonn
%O 1,2
%A _Julio Cesar Hernandez-Castro_, Jun 29 2011
%E 0 added by _Arkadiusz Wesolowski_, Jun 29 2011
%E a(9) from _Arkadiusz Wesolowski_, Jul 02 2011
%E a(10) from _Charles R Greathouse IV_, Oct 14 2011
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