cp's OEIS Frontend

A067834 Numbers k such that in the ring Z[sqrt(3)] the norm of (-1+sqrt(3))^k-1 is prime.

%I A067834 #36 Dec 05 2024 09:26:43
%S A067834 2,3,7,13,19,43,61,151,257,751,859,1453,3767,3889,8171,15959,21499,
%T A067834 22679,23297,31277,43609,57037,61961,103087,115931,173647,215959,
%U A067834 496073
%N A067834 Numbers k such that in the ring Z[sqrt(3)] the norm of (-1+sqrt(3))^k-1 is prime.
%C A067834 The definition implies that k must be a prime.
%C A067834 The norm of (-1+sqrt(3))^k-1 is given by (-2)^k - lucasV(-2, -2, k)+1, where lucasV(-2, -2, k) is the solution of the recurrence relation v(n) = -2*v(n-1) + 2*v(n-2) with v(0)=2 and v(1)=-2.
%C A067834 a(29) > 517000. - _Serge Batalov_, Oct 24 2024
%H A067834 Mike Oakes, <a href="http://groups.yahoo.com/group/primenumbers/message/5217">Posting to primenumbers list on Feb 08 2002</a>
%H A067834 Mike Oakes, <a href="/A067834/a067834.txt">4 new forms of primes</a>, digest of 4 messages in primenumbers Yahoo group, Feb 8 - Feb 11, 2002.
%e A067834 3 is a term because (-2)^3-lucasV(-2,-2,3)+1 = -8-(-20)+1 = 13 and 13 is prime.
%t A067834 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 *)
%o A067834 (PARI) isok(n)={ispseudoprime(([0, 1; 2, 2]^n*[2; 2])[1, 1] - 2^n - (-1)^n)} \\ _Andrew Howroyd_, Oct 24 2024
%Y A067834 Cf. A080040.
%K A067834 nonn,nice,more
%O A067834 1,1
%A A067834 _Mike Oakes_, Feb 09 2002
%E A067834 Corrected and extended by _Aurelien Gibier_, Oct 24 2024
%E A067834 a(28) from _Serge Batalov_, Oct 24 2024