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Numbers k such that 3^k + k is prime.
3

%I #18 Mar 01 2024 14:17:37

%S 2,8,34,1532,18248

%N Numbers k such that 3^k + k is prime.

%C Note that if n > 2 and n+1 is prime then (by Fermat's theorem) n+1 divides 3^n+n.

%C If it exists, a(6) > 100000. - _Hugo Pfoertner_, Mar 01 2024

%t Do[ If[ PrimeQ[ 3^n + n ], Print[ n ] ], {n, 0, 3000} ]

%t v={2}; Do[If[EvenQ[n]&&Mod[n, 3]!=0&&!PrimeQ[n+1]&&PrimeQ[3^n+n], v=Append[v, n]; Print[v]], {n, 3, 19000}]

%t Select[Range[18500],PrimeQ[3^#+#]&] (* _Harvey P. Dale_, Jul 23 2013 *)

%o (PARI) is(n)=ispseudoprime(3^n+n) \\ _Charles R Greathouse IV_, May 22 2017

%Y Cf. A104743, A052007, A057909.

%K nonn,hard,more

%O 1,1

%A _Robert G. Wilson v_, Nov 16 2000

%E 18248 from _Farideh Firoozbakht_, Aug 21 2003