%I #10 Nov 18 2018 10:37:00

%S 2,2,3,2,2,3,2,2,11,2,5,11,2,2,5,71,2,3,2,2,167,2,17,3,2,197,149,2,2,

%T 3,3,2,2267,2,2,3,3,2,29,2,2531,167,2,7,3,3,2,61,2,2,11,2,2,157,2,5,7,

%U 7,149,3,5,2,379,2,41,3,2,2,3,79,11,3,2,2,97,3,2,3,3,2,1321,2,17,31,2,61

%N Least number k > 0 such that ((2n+1)^k - 2^k)/(2n-1) is prime.

%C All terms are primes.

%C a(n) = 2 for n = {1,2,4,5,7,8,10,13,14,17,19,20,22,...} = A067076 Numbers n such that 2n+3 is a prime.

%C a(34),...,a(40) = {2,2,3,3,2,29,2}.

%C a(42),...,a(80) = {167,2,7,3,3,2,61,2,2,11,2,2,157,2,5,7,7,149,3,5,2,379,2,41,3,2,2,3,79,11,3,2,2,97,3,2,3,3,2}.

%C a(82),...,a(90) = {2,17,31,2,61,7,2,2,5}.

%C a(93),...,a(95) = {383,2,2}.

%C a(97),...,a(100) = {2,2,5,7}.

%C a(102),...,a(124) = {13,11,2,5,5,17,3,103,2,19,2,2,3,2,31,37,2,2,3,3,7,3,2}.

%C a(127),...,a(131) = {2,61,31,2,157}.

%C a(133),...,a(142) = {2,2,7,3,2,13,2,2,7,3}.

%C a(144),...,a(146) = {173,2,11}.

%C a(148),...,a(150) = {3,17,107}.

%C a(n) is currently unknown for n = {33,41,81,91,92,96,101,125,126,132,143,147,...}.

%t Do[k = 1; While[ !PrimeQ[((2n+1)^k - 2^k)/(2n-1)], k++ ]; Print[k], {n, 100}] (* _Ryan Propper_, Mar 29 2007 *)

%t lnk[n_]:=Module[{k=1},While[!PrimeQ[((2n+1)^k-2^k)/(2n-1)],k++];k]; Array[ lnk,90] (* _Harvey P. Dale_, May 19 2012 *)

%Y Cf. A067076.

%Y Cf. A000043 = Primes p such that 2^p - 1 is prime.

%Y Cf. A001348 = Mersenne numbers: 2^p - 1, where p is prime.

%Y Cf. A057468 = numbers n such that 3^n - 2^n is prime.

%Y Cf. A125958 = Least number k > 0 such that (2^k + (2n-1)^k)/(2n+1) is prime.

%K hard,nonn

%O 0,1

%A _Alexander Adamchuk_, Feb 07 2007

%E More terms from _Ryan Propper_, Mar 29 2007