A067834 Norm in ring Z[sqrt(3)] of (((-1+sqrt(3))^n)-1) is prime.

2, 3, 7, 13, 19, 43, 61, 151, 257, 751, 859, 1453, 3767, 3889, 8171, 15959, 61961

Offset

1,1

Links

Table of n, a(n) for n=1..17.

Mike Oakes, Posting to primenumbers list on Feb 08 2002

Mike Oakes, 4 new forms of primes, digest of 4 messages in primenumbers Yahoo group, Feb 8 - Feb 11, 2002.

Formula

(-2)^n-lucasV(-2, -2, n)+1, where lucasV(-2, -2, n) is the solution of the recurrence relation v[0]=2, v[1]=-2, v[n+2]=-2*v[n+1]+2*v[n] n >= 0.

Example

a(2)=3 because (-2)^3-lucasV(-2,-2,3)+1 = -8-(-20)+1 = 13 and 13 is prime.

Mathematica

v[0] = 2; v[1] = -2; v[n_] := v[n] = -2*v[n-1] + 2*v[n-2] ; s = {}; Do[If[PrimeQ[(-2)^n - v[n] + 1], Print[n]; AppendTo[s, n]], {n, 8171}]; s (* Jean-François Alcover, Apr 18 2011 *)

Crossrefs

Sequence in context: A249749 A171817 A156300 * A070754 A325875 A049887

Adjacent sequences: A067831 A067832 A067833 * A067835 A067836 A067837

Keyword

nonn,nice,more

Author

Mike Oakes, Feb 09 2002

Status

approved