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Least nonnegative k such that 3^(2^n)+k is prime.
1

%I #26 Nov 02 2024 10:53:23

%S 0,2,2,2,26,70,92,190,788,436,86,3032,13622,2810,7562,33172,16942

%N Least nonnegative k such that 3^(2^n)+k is prime.

%C The associated primes are in A157980.

%C All 3^(2^n)+a(n) = A157980(n) for n <= 11 are certified primes. - _D. S. McNeil_, Mar 15 2009

%H Florentin Smarandache, <a href="http://arxiv.org/abs/0903.1380">Seven Conjectures in Geometry and Number Theory</a>, arXiv:0903.1380 [math.GM], Mar 8, 2009.

%F a(n) = Min{k>=0 such that 3^(2^n)+k is prime} = Min{k>=0 such that A000244(A000079(n))+k is in A000040}.

%e a(0) = 0 because 3^2^0 + 0 = 3^1 + 0 = 3 is prime. a(1) = 2 because 3^2^1 + 2 = 3^2 + 0 = 3 is prime. a(2) = 2 because 3^4 + 2 = 83 is prime. a(3) = 2 because 3^8 + 2 = 6563 is prime. a(4) = 26 because 3^16 + 26 = 43046747 is prime. a(5) = 70 because 3^32 + 2 = 1853020188851911 is prime. a(6) = 92 because 3^64 + 2 = 3433683820292512484657849089373 is prime.

%t lnnk[n_]:=With[{c=3^(2^n)},NextPrime[c]-c]; Join[{0},Array[lnnk,10]] (* The program generates the first 11 terms of the sequence. *) (* _Harvey P. Dale_, Nov 02 2024 *)

%o (PARI) { a(n) = nextprime( 3^(2^n) ) - 3^(2^n) } \\ _Max Alekseyev_, Sep 13 2009

%Y Cf. A000040, A000079, A000244, A000215, A157980.

%K more,nonn,hard

%O 0,2

%A _Jonathan Vos Post_, Mar 10 2009

%E a(7)-a(11) from _D. S. McNeil_, Mar 15 2009

%E a(12)-a(14) from _Max Alekseyev_, Sep 13 2009

%E a(15)-a(16) from _Donovan Johnson_, Jul 11 2010