A125954 Least number k > 0 such that ((2n+1)^k - 2^k)/(2n-1) is prime.

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, 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, 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

Offset: 0

Author

Alexander Adamchuk, Feb 07 2007

Keywords

• hard
• nonn

Comments

All terms are primes.

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.

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

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}.

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

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

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

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}.

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

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

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

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

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

Crossrefs

Cf. A067076.

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

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

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

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

Programs

Mathematica

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

Extensions

More terms from Ryan Propper, Mar 29 2007